Download TI-92 Geometry Tour

TI-92 Geometry Tour
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The following materials were produced by the Glenbrook South High School mathematics department and may not be
reproduced without permission (phone: 847-486-4683 or e-mail: [email protected]).
The following tutorials and reference material are to help you learn geometry through a dynamic
environment. This means you can interact and manipulate with geometric objects without having to
redraw them. The geometry program on the TI-92 calculator will accompany your textbook, UCSMP
Geometry, and will be used throughout the school year to help you conjecture and learn mathematical
ideas and principles.
The lessons in this guide are established through a building process. In each subsequent lesson, you
will be asked to recall how you created an object from a previous activity. Therefore, it might be
difficult to jump around between lessons. If you work through all of the lessons, and forget how to
create something, use the Index in the back to help guide you to the exploration where it was first
explained. Keep this handout for future reference.
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TABLE OF CONTENTS
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EXPLORATION TITLE
0-1
0-2
0-3
0-4
0-5
0-6
0-7
0-8
0-9
0-10
Hey, my screen isn't dark enough (too light)! .................................
Oops! I pressed the wrong key ........................................................
Create a new folder .........................................................................
Create a new document ...................................................................
Create, label, and move points ........................................................
Correct an error or clear the screen .................................................
Create two intersecting lines ...........................................................
Create, measure, and change a circle ..............................................
Create, measure, and change an angle ............................................
Create, measure, and change a triangle ...........................................
PAGE(S)
5
5
5
5-6
6-7
7-8
8
9-10
10
11
1-3 Equation of lines ............................................................................. 12-13
1-5 Drawing in perspective ................................................................... 13-14
2-4 Midpoints ........................................................................................ 15
2-7 Triangle inequality .......................................................................... 15-16
2-8 Conjectures ................................................................
16-17
3-1
3-2
3-3
3-6
3-7
3-8
3-8P
Angle bisector .................................................................................
Rotations ...................................................................
Vertical angles.................................................................................
Slope and parallel lines ...................................................................
Perpendicular lines ..........................................................................
Perpendicular bisector .....................................................................
Find the hidden treasure ..................................................................
18
19
20
20-21
22
22-23
23-24
4-1
4-2
4-3
4-4
4-5
4-6
4-7
Reflecting points .............................................................................
Reflecting figures in a coordinate plane..........................................
Reflections and minimum distance .................................................
Translations: Composing reflections over parallel lines .................
Rotations: Composing reflections over intersecting lines...............
Translations and vectors .................................................................
Glide reflections ..............................................................................
24
25
26-27
27-28
28-29
29-30
30
5-2
5-4
5-5
5-5P
5-7
Congruence and equality .................................................................
Alternate interior angles ..................................................................
Perpendicular bisector theorem .......................................................
Capture the flag ...............................................................................
Sum of angles in a polygon .............................................................
31
31
32
32-33
33-34
6-1 Reflection-symmetric figures .......................................................... 34-35
6-2 Isosceles and equilateral triangles ................................................... 35-36
6-2P Shark attack ..................................................................................... 36-37
6-3 Constructing parallelograms ........................................................... 37-38
6-4 Constructing a kite .......................................................................... 38-39
6-5 Constructing a trapezoid ................................................................. 39-40
6-M Mystery quadrilaterals ..................................................................... 40-42
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6-6 Rotation symmetry .......................................................................... 43-44
6-7 Regular
polygons
The following materials
were produced
by the............................................................................
Glenbrook South High School mathematics department and44-45
may not be
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TABLE OF CONTENTS (CONTINUED)
EXPLORATION TITLE
PAGE(S)
7-2
7-5
7-6
7-7
7-7P
7-9
Congruent triangles .........................................................................
SsA condition and HL congruence .................................................
Tessellations ....................................................................................
Properties of parallelograms ...........................................................
Bouncing off the walls ....................................................................
Exterior angles ................................................................................
45-47
47-48
49-50
50
51
52
8-1
8-2
8-2P
8-3
8-4
8-4P
8-5
8-6
8-7
8-8
8-8P
Perimeter of a regular polygon ........................................................
Area of a rectangle ..........................................................................
Optimal quadrilaterals .....................................................................
Areas of irregular polygons .............................................................
Area of a triangle.............................................................................
Triangle in a rectangle.....................................................................
Area of trapezoids ...........................................................................
Pythagorean Theorem .....................................................................
Circumference and arc length .........................................................
Area of a circle through data analysis .............................................
Area of tangent circles ....................................................................
53
53-54
54-55
55-56
56-58
58-59
59-61
61-62
62-63
64-66
66-67
9-5 Famous paths in geometry...............................................
68-69
10-6 Creating a toolbar to reference formulas ......................................... 70-71
11-6
11-7
11-8
11-8P
Distance formula .............................................................................
Equations of circles .........................................................................
Means and midpoints ......................................................................
Midpoints and areas ........................................................................
72-73
73-75
75-76
77-78
12-1
12-2
12-3
12-4
12-5
12-6
The transformation sk......................................................................
Size changes ....................................................................................
Properties of size changes ...............................................................
Proportions ......................................................................................
Similarity.........................................................................................
The fundamental theorem of similarity...........................................
78-79
80
81
82
82-83
84
13-1
13-2
13-2P
13-3
The SSS similarity theorem ............................................................
The AA and SAS similarity theorems.............................................
Triangles in a trapezoid ...................................................................
The side splitting theorem ...............................................................
84-85
85-86
86-87
87
Index ............................................................................................... 88-91
Variable Index ................................................................................. 92-93
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INTRODUCTION TO THE TI-92
Objective: In this activity, you will practice some of the basic functions of the geometry on the TI-92
calculator, including use of the Pointer, Point, Line, Measure, and Construct toolbars.
EXPLORATION 0-1. HEY, MY SCREEN ISN'T DARK ENOUGH (TOO LIGHT)!
Turn the calculator ON . ◊ and + will darken your screen one notch if necessary. ◊
and
- will lighten your screen one notch if necessary. Repeat until you reach the desired contrast. And
once you’ve gotten your screen adjusted the way you like it, you can reduce glare by snapping the lid to
the calculator. You may find it easier not to do this, however, and that’s fine, too.
EXPLORATION 0-2. OOPS! I PRESSED THE WRONG KEY
There are a lot of keys in this small hand-held computer (large calculator). Therefore, it is natural to
press the wrong key at the worst possible moment (that's called Murphy's Law). When you press the
wrong key, press the ESC button to get you back to where you started. Also, press ESC when
you want to return to the pointer (cross). Get in the habit of pressing the escape key when you’re
done with a certain feature.
EXPLORATION 0-3. CREATE A NEW FOLDER
First, you need to create a folder to save all of the work you do in this course.
Press 2nd – (we’re going into the VAR-LINK menu) , F1 Manage
and 5:Create Folder. Type
your name in the Folder: box and press ENTER twice. A folder with your name should now appear
in the directory. You should not need to do this again for the remainder of the year.
EXPLORATION 0-4. CREATE A NEW GEOMETRY DOCUMENT
Start the geometry program by pressing APPS , and choose 8:Geometry » 3:New... to begin. When
you select Geometry and New you can either press the corresponding number (8 and then 3) or you can
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arrow down using the arrow pad (blue wheel). Use the arrow pad to arrow right and arrow down to
your name and press ENTER to change the Folder: to your name. Arrow down (do not press enter)
and type points in the Variable: box and press ENTER twice. You should now see a pointer (cross)
at the center of the screen and a bunch of tools to represent menus at the top. Use this process each
time you want to create a new geometry file. You may have to refer back to this page until you get
comfortable with it.
Now take a minute to explore the different function keys, or F-keys, like F1, F2, . . . Don’t actually
select any particular function. Just select each function key and see what options are in that menu.
EXPLORATION 0-5. CREATE, LABEL, AND MOVE POINTS.
Press F2 and choose 1:Point. Note the cross has changed to a pencil, meaning that you are ready to
create points on the screen. Use the arrow pad to move the pencil to any location on the screen and
press ENTER . Immediately type A to name the point. To make capital letters (which is customary
for points) just hit  which is right above ON . This key acts as a shift key. 
It is best to immediately type the letter before another point is created. If you keep creating points, it’s
much more difficult to label the points. Use the arrow pad to move the pencil away from the point so
you can see it on the screen. Press ENTER again to make another point and immediately type B to
name it. Repeat this process until you have made five points named A through E.
If you forget to label or mislabeled a point, first press ESC to get out of the point making function.
Your calculator can’t read your find and will assume you’re making more points until you tell it
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otherwise. After hitting ESC you should see the pointer (cross), move the arrow pad over to the point
until the screen reads THIS POINT, press F7
and choose 4:Label, press ENTER , use the left
arrow key (2 keys to right of space bar) to delete the old label (if there was one), type in or replace the
label, and press ESC .
Press ESC
if the pointer (cross) is not on the screen. Use the arrow pad and move on top of point c
until the screen reads THIS POINT. Hold down the HAND TOOL with your left thumb and move
point c around the screen by using the arrow pad with your right thumb.
Note: The HAND TOOL and arrow pad (blue wheel) together function like a mouse. Just like when you click and
drag something on the computer with a mouse you must hold the button down, on the TI-92 you must hold the
HAND TOOL button down while you use the arrow pad for the movement.
Now let's just reposition (or drag) the label. Use
the arrow pad and move on top of the label for
point c until the screen reads THIS LABEL.
Hold down the HAND TOOL with your left
thumb and move the label for point c with the
arrow pad .
Move the points and labels around until your
screen matches the one shown to the right.
When you move a point the label follows.
EXPLORATION 0-6. CLEAR SCREEN/DRAW SEGMENT/CORRECT AN ERROR
Let's start over with a fresh screen by pressing F8 and choosing 8: Clear All. The calculator will
ask you to press enter to confirm this request.
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Next draw a segment anywhere on the screen by
pressing F2 and 5: Segment. Notice that the
icon in the "toolbar" across the top of the screen
for F2 has changed from a point to a segment.
Use the arrow pad to move to where you want the
segment to start, press ENTER -- don't forget
to type a letter immediately after so you don't
have to use the label command. Then repeat for
the other endpoint and type a label.
The following information is needed when you make an error. If you press the wrong key, press
ESC to clear the command. Let's practice deleting things. First press ESC to get back to the
pointer and to tell the calculator that you're no longer interested in making segments. Move the key
pad on top of the object until it says THIS SEGMENT (it won't say this if you point at the endpoints -- you need
to move inbetween the points somewhere), press ENTER (it becomes dotted) , press F8 and choose
7:Delete (or press the LEFT ARROW). Notice that the points are still on the screen. You could delete
each separately, but instead select a point and then hold down the shift key (the UP ARROW).
EXPLORATION 0-7. CREATE TWO INTERSECTING LINES
CREATE A NEW DOCUMENT (even though your screen is blank -- you need to practice) with the
variable lines. Refer to Step 4 if you forgot. Get in the habit of 1. referring back when you forget
something, then 2. ask a group member, then 3. ask the teacher if needed and I'll be glad to help.
A line can be created with two points. Select F2
and choose 4:Line. Move the pencil to the lower-
right side of the screen and press ENTER . Label this point A. Move the pencil to the upper-left side
of the screen so that the line is on a diagonal and press ENTER . A line like this, which is neither
horizontal nor vertical, is called oblique. Create another line so that the two lines create a large X as
shown to the lower-left. Label the point on the second line B.
Press F2
and choose 3:Intersection Point. Move the pencil to the intersection point until the
message POINT AT THIS INTERSECTION appears. Press
ENTER , label this point C, and
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ESC
to return to the pointer. Move the cross to the upper-right hand portion of the screen until the
screen reads THIS LINE. Press F2
and choose 2:Point on Object, press ENTER , label this
new point D (be careful not to label the line), then ESC . Your screen should match the picture
shown above to the right.
EXPLORATION 0-8. CREATE, MEASURE, AND CHANGE A CIRCLE
CREATE A NEW DOCUMENT with the variable circ.
A circle is formed by selecting a center and a
point on the circle. The radius is the distance
from the center of the circle to any point on the
circle. Press F3 and choose 1:Circle, press
ENTER to mark the circle's center and type C
for center, and move the pencil outward to
establish the size of the circle's radius. Make a
circle large enough to cover one-fourth of the
screen and press ENTER to finish the circle.
Construct a radius by pressing F2 & choosing 5:Segment, move the pencil on the circle's center,
press ENTER and then drag the segment to a point on the circle until the screen reads ON THIS
CIRCLE, and press ENTER
and type A. Press ESC
to return to the pointer.
Measure the radius of the circle by pressing F6 and choosing 1:Distance & Length, move the
pencil to the middle of the segment until the screen reads LENGTH OF THIS SEGMENT, and press
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ENTER . Press ESC
press ENTER
to return to the pointer, move the pencil on top of the radius measurement,
to select (dotted), and arrow left
to delete the measurement.
Another method to measure the length of a segment is to find the distance between the endpoints by
moving the pencil to one endpoint, pressing
ENTER , moving to the other endpoint, and pressing
ENTER . Press ESC
to return to the pointer, drag the measurement to the upper right-hand
portion of the screen, press
measurement, press ENTER
Consult the diagram below.
F7
and choose 5:Comment, drag the pencil to the left of the
, type radius= , hit ESC , and then drag the comment as needed.
The circumference of the circle is the length around the circle. Measure the circumference by pressing
F6 and choosing 1:Distance & Length, move the pencil to the circle until the screen reads
CIRCUMFERENCE OF THIS CIRCLE, and press ENTER . Type a comment (abbreviate with
circum if you wish) for the measurement and drag it underneath the radius measurement .
The area of the circle is the amount of space inside the circle. Measure the area of the circle by
pressing F6 and choosing 2:Area, move the pencil to the circle until the screen reads THIS
CIRCLE, and press ENTER . Comment (area=) and drag this measurement underneath the other
measurements. Drag the circle to the left-hand side of the screen by moving its center as shown below
to the left.
You can change the size of the circle by moving the pointer on top of the circle, pressing ENTER ,
and dragging with the HAND TOOL toward or away from the circle's center. Modify the
measurements on your screen until they match the picture shown to the right above (within 0.1 cm and
0.3 cm2).
EXPLORATION 0-9. CREATE, MEASURE, AND CHANGE AN ANGLE
CREATE A NEW DOCUMENT with the variable angles.
An angle can be created with two rays. Press F2
and choose 6:Ray. Move the pencil to the left-
hand-center part of the screen and press ENTER . Label this vertex point B and direct the ray to the
upper right-hand portion of the screen and press ENTER . Move the pencil on top of point B until
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the screen reads THIS POINT, press ENTER
to make another ray, and direct the second ray to the
lower right-hand portion of the screen and press ENTER . Create point A on the top ray and point C
on the bottom ray by using F2
Use the HAND TOOL
and 2:Point on Object as shown in the diagram below to the left.
to move points A and C close to the vertex of the angle.
An angle can be measured if a point on one ray, the vertex, and a point on the other ray are selected.
Press F6 and choose 3:Angle. Move the pencil over point A and press ENTER , over point B and
press ENTER , over point C and press ENTER . The measure of <ABC should appear on the
screen. Press ESC
to return to the pointer and move the pencil along ray BC until the screen reads
THIS RAY. Hold down the HAND TOOL and use the key pad to change the angle until it almost
equals 65˚ (within 0.5˚). If necessary, move the angle measurement inside the angle, and the points on
the angle closer to the vertex so they remain on the screen as shown in the picture to the top-right.
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EXPLORATION 0-10. CREATE, MEASURE, AND CHANGE A TRIANGLE
CREATE A NEW DOCUMENT with the variable triangle.
Press F3 and choose 3:Triangle. Press ENTER and label the first point a, drag the pencil and
repeat the process until you have created ∆abc. Use the procedures in Steps 8 and 9 to measure the
triangle's perimeter, area, segment lengths, and angle measurements. Organize the measurements as
shown in the image below to the left.
It is possible to make calculations on the screen to explore geometric patterns. Press F6 and choose
6:Calculate. A calculation strip with a cursor will appear at the bottom of the screen. Using the arrow
pad (blue wheel) arrow up until one of the side length measurements is highlighted, press ENTER
and an "a" should appear on the calculation strip (the number has been stored in what the calculator calls
"a"),type a + since we want to add it to the other sides, highlight another side length measurement by
arrowing up, press ENTER
and a "b" should appear on the calculation strip, + , highlight the last
side length measurement, press ENTER
and a "c" should appear on the calculation strip. Press
ENTER to execute the calculation and a value next to an R: should be on the screen as shown above
to the right.
Press ESC to return to the pointer and drag one of the vertices of the triangle around the screen and
observe the changes in the numbers and the similarity between this number and one of your
measurements. Explain what you have found below.
Find the sum of the angles in a triangle. Drag the vertices around to modify the angle measurements.
Explain what happens to this sum. Why do you think this is happening? Explain below.
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EXPLORATION 1-3. EQUATION OF LINES.
Objective: Distinguish the equations of vertical, horizontal, and oblique lines.
1. CREATE A NEW DOCUMENT with the variable eqline.
2. Create a coordinate plane
Press F8 and choose 9:Format, change the Coordinate Axes to RECTANGULAR and the Grid to
ON, press
ENTER
to save, move the cross on top of the unit on the x-axis (probably 0.5) until
the screen reads THIS UNIT, hold the HAND TOOL , and drag the unit until it changes to 1 as
shown in the picture below to the left.
3. Place points at (3,2) and (-1,-1). Create a line which passes thru both of these points. Make sure the
screen reads THRU THIS POINT before pressing ENTER , as shown in the diagram above to the
right.
4. Find the equation of the line in slope-intercept form
Press F6 and choose 5:Equation & Coordinates, move the pencil on top of the line until the screen
reads EQUATION OF THIS LINE, press ENTER and the equation should appear next to the
pencil in slope-intercept form. Write the equation of the line below, and state the slope and y-intercept.
EQUATION: ______________________
SLOPE: ________
Y-INTERCEPT: ___________
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5. Press ESC to return to the pointer. Drag the point (3,2) to the point (3,-1). Write the equation of
the line below, and state the slope and y-intercept.
EQUATION: ______________________
SLOPE: ________
Y-INTERCEPT: ___________
6. Drag the point (3,-1) to the point (-1,2). Write the equation of the line below, and state the slope and
y-intercept.
EQUATION: ______________________
SLOPE: ________
Y-INTERCEPT: ___________
7. Explain below how you can distinguish vertical, horizontal, and oblique lines by their equation.
8. Use your explanation in step 7 to determine whether the following equations represent vertical,
horizontal, or oblique lines. Use your calculator only to CHECK your answer.
Equation
a. x = 2
Your Answer Passes Thru (to check on calculator)
_____________
(2,-1), (2,2)
b. y = -3
_____________
1
1
c. y = 2 x - 2 _____________
(0,-3), (-4,-3)
(-1,-1), (1,0)
EXPLORATION 1-5. DRAWING IN PERSPECTIVE
Objective: In this activity, you will create a box in perspective where you can manipulative the
vanishing point to see different viewpoints. Use page 29 of your text as a reference.
1. CREATE A NEW DOCUMENT with the variable perspect.
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2. Create a square
Press F3 and choosing 5:Regular Polygon,
move the pencil near the center of the screen,
press ENTER to locate the center of the
square, move the pencil about 2 cm away from
the center and press ENTER , move the pencil
away from the point on the dotted circle until the
screen reads 4 (sided polygon) and press
ENTER as shown to the right. A square
should now appear on your screen.
3. Place a point on your screen to represent the
vanishing point as shown in page 29 on your text.
4. Connect segments from the vanishing point to
three of the vertices of the square as shown in the
figure at the right. Make sure the calculator reads
THIS POINT before pressing ENTER .
5. Create parallel lines to the sides of the square which connect to the vanishing point.
Press F4 and choose 2: Parallel Line, move the pencil to the right-hand side of the square until the
screen reads PARALLEL TO THIS SIDE OF THE POLYGON, press ENTER ,
move the pencil on top of the lowest segment
until the screen reads ON THIS SEGMENT and
press ENTER . The parallel line should now
appear on the screen. Create a point of
intersection between the parallel line and the
middle perspective line. Now create another
parallel line thru the point of intersection and
parallel to the top segment of the square as shown
in the diagram to the right. Create an intersection
point between the new parallel line and the leftmost perspective segment.
6. Create five overlapping segments which create the three dimensional perspective of the box.
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7. Hide everything except the box and vanishing
point. Press F7 and choose 1:Hide/Show,
move the pointer on each object you wish to hide
and press ENTER . The object(s) you are
hiding should become dotted. When you are
finished hiding, press ESC to return to the
pointer. If you make a mistake, undo, and try
again. Your screen should resemble the picture to
the right.
8. From the current perspective, you can view three faces of the box. Move the vanishing point around
the screen and observe the changing perspectives of your box.
Where is the vanishing point when the top face is not showing?
Where is the vanishing point when the side face is not showing?
Where is the vanishing point when the top and side face are not showing?
EXPLORATION 2-4. MIDPOINTS
Objective: Determine how a midpoint relates to and affects a line segment.
1. CREATE A NEW DOCUMENT with the variable midpoint.
2. Place an oblique segment across 3/4 of the screen. Label the endpoints d and g.
3. Create a midpoint on a segment
Press F4 and choose 3:Midpoint, place the pencil on top of the segment until the screen reads
midpoint of this segment, press ENTER
and label the midpoint o.
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4. Measure the length of segments do, og, and dg.
Label the measurements and organize them on the
screen as shown to the right.
5. Drag point d around the screen and observe
the changes in your measurements. Make two
conjectures below about the effect of a midpoint
on a segment.
6. Clear everything from the screen by pressing
F8 and 8:Clear All. Press ENTER to
confirm.
7. Create a large circle in the center of the screen.
Label the center u. Construct a line which passes
thru the circle and the center of the circle. Make
a point of intersection with the circle and the line.
Create an overlapping segment connecting the
two points shared by the line and the circle. This
segment which has endpoints on the circle and
passes thru the center of the circle is called the
diameter of the circle. Label the diameter
endpoints c and b. Hide the line so your diagram
is similar to the one shown to the right.
8. Measure the radii (segments cu and ub). Drag and modify the size of the circle and observe changes
in your measurements. Explain below how the center, radii, and diameter of the circle relate to the
concept discussed earlier this exploration.
EXPLORATION 2-7. TRIANGLE INEQUALITY
Objective: Determine how the lengths of the side of a triangle relate to each other.
1. CREATE A NEW DOCUMENT with the
variable inequal.
2. Create a large triangle ( F3 , 3) which covers
about one-half of the screen. (Remember to type the
label after each time you press enter to make each of the
three vertices.) Measure the length of each of the
sides of the triangle and label each of the
measurements.
3. Calculate the og + gp, og + op, op + gp, label
and organize the calculations as show to the right.
4. Compare the sum of two sides of the triangle to the third side. Record and organize the information
below.
sum of two sides
og + gp =
og + op =
p. 19
op + gp =
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third side
op =
gp =
og =
How do the sums compare to the 3rd side? Which one is greater?
Move point o close to segment gp. Record the data below.
sum of two sides
third side
og + gp =
op =
og + op =
gp =
op + gp =
og =
How do the sums compare to the 3rd side? Which one is greater?
Move point o on top of segment gp so the triangle turns into a segment. Record the data below.
sum of two sides
third side
og + gp =
op =
og + op =
gp =
op + gp =
og =
How do the sums compare to the 3rd side? Which one is greater?
In a triangle, make a conjecture about the information above using an if...then... statement.
EXPLORATION 2-8. CONJECTURES
Objective: Determine the validity of conjectures
1. CREATE A NEW DOCUMENT with the variable conject.
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2. Create a large triangle. Construct the
midpoint of each side and connect them with
three segments as shown.
3. Test the following conjecture: If the midpoints
of two sides of a triangle are joined, the segment
is parallel to the third side.
At this point, do you feel this conjecture is true?
How sure are you?
To discern with more certainty whether this is
true, press F6 and choose 8: Check Propery.
Arrow right to choose Parallel from the submenu. Let's check if ED // BC first. Arrow over
to ED (you'll see IS THIS SEGMENT), press
ENT , and select BC (you'll see PARALLEL
TO THIS SIDE OF THE TRIANGLE). Now
press enter one final time. Are the segments
parallel?
Now drag around one of the vertices of the original triangle by pressing ESC , moving to a point, and
pressing the hand tool. Do you find that this conjecture is true in general?
4. Test the following conjecture: Area(∆ABC) = 4 • Area(∆DEF) or, in other words, the ratio of their
two areas is 4. At this point, do you feel this conjecture is true? How sure are you?
Have the calculator find the areas of the triangles. However, recall that we made ∆DEF by joining
three segments. The calculator cannot find the area of a region that was not constructed as a polygon.
Therefore, we must make an overlapping triangle on top of the existing segments. Create ∆DEF and
then you will be able to find its area. Drag the measurements to the upper right hand portion of the
screen.
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Next we will use the 6: Calculate tool under the
F6 menu to divide the larger area of ∆ABC by
the smaller area of ∆DEF. If you've forgotten
how to do this, refer back to Exploration 0-10.
Remember that you have an index! I won't
usually tell you where to refer -- you're expected
to use the index or ask a group member.
In your calculation strip across the bottom you
should see a/b (you only type the / which is
÷ on the calculator, the 92 does the letters when
you highlight a number and press enter).
Press ESC since we no longer wish to calculate. Get in the habit of pressing escape when you're
done with something! Now drag around one of the vertices of the original triangle. Be sure to move
around in all directions. Do you find that this conjecture is true in general?
EXPLORATION 3-1. ANGLE BISECTOR
Objective: Determine how an angle bisector relates to and affects an angle.
1. CREATE A NEW DOCUMENT with the variable angbis.
2. Create an angle and label it so it reads < TAL. (the symbol is in the catalog, so it is easier to just use the
"less than" symbol for angles. The < is typed by pressing 2nd and 0 .) Measure and label (comment) the
angle. See EXPLORATION 0-9 if you forgot.
3. Create an angle bisector
Press F4 and choose 5:Angle Bisector, move
the pencil on top of each point IN ORDER, and
press ENTER after selecting each point. The
angle bisector should divide  TAL into two
smaller angles. Place a point on the angle
bisector and label it C as shown to the right.
4. Measure and label the two smaller angles,
TAC and CAL.
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5. Drag one of the rays around the screen so the angles change measurement, and observe the changes
in your measurements. Make two conjectures below about the effect of an angle bisector on an angle.
6. Construct a triangle and the angle bisectors of all of the angles. Complete the following conjecture:
If the angles of a triangle are bisected, then ...
7. What side of a triangle is closest to the intersection of the angle bisectors? Explain how you know.
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EXPLORATION 3-2. ROTATIONS
Objective: Determine how magnitude and sign of a rotation affects the preimage
1. CREATE A NEW DOCUMENT with the variable rotate.
2. Create a small, pointy triangle as illustrated.
Create a point, O, near the triangle. To create the
numerical value of the rotation, use F7 and
choose 6: Numerical Edit. Press ENT to get
the edit box and type 90. Press ESC .
To rotate the triangle 90° in the positive direction,
select F5 and 2: Rotation. Select the triangle
to rotate, then point O as the center of rotation,
and finally select the 90, and presto!
3. With a 90° rotation, how much did the triangle turn? Use answers such as quarter-turn, half-turn, threequarters turn, etc.
Make a sketch of the preimage and image in the box below. Label the preimage Start and label the
image 90 in your sketch.
SKETCHES
.
4. Undo the 90° rotation by hitting F8
and
choose D:Undo or press ◊ and z . Change
the 90 to 180 by arrowing to the number, pressing
ENT twice (once to select and once to get the edit box
to appear) and then
Error!
Sketch the resulting image triangle in the same
drawing as the 90° rotation. Label this triangle
180°.
Repeat these steps for a 270°, 360°, and 450° rotation. Answer the questions below and make a sketch.
With a 270° rotation, how much did the triangle turn?
With a 360° rotation, how much did the triangle turn?
With a 450° rotation, how much did the triangle turn?
4. What happens with a negative rotation? Try a numerical edit of -45°. Be sure you use the gray
(–) key for the negative. Describe the difference between negative and positive rotations.
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EXPLORATION 3-3. VERTICAL ANGLES
Objective: Determine the relationship between two vertical angles.
1. CREATE A NEW DOCUMENT with the
variable vertang.
2. Create two lines intersecting near the middle of
the screen. Find a point of intersection, create
and label points on the lines so they correspond
with the diagram to the right.
3. Measure vertical angles DAM and GAO.
Repeat this procedure with vertical angles GAM
and OAD. Drag around the lines so the angle
measurement change.
4. Make a conjecture below about your findings in step 3.
EXPLORATION 3-6. SLOPE AND PARALLEL LINES.
Objective: Verify the slope formula and determine some properties which relate to parallel lines.
1. CREATE A NEW DOCUMENT with the variable parallel.
2. Use the Format tool in F8 to turn on the coordinate axes to a RECTANGULAR grid. Turn the
grid to ON. Press ENT to save these changes. Change the unit label 0.5on the axes to equal 1 by
first getting the prompt THIS UNIT. This is a little tricky. Move around until you get it. You don't
want THESE AXES but the calculator will try to get you to bite. Don't. Once it says THIS UNIT (as
shown below to left), slowly drag it left to get a grid like the one to the right.
3. Create a line passing thru (-1,-1), and (1,0). Label the line b after pressing ENTER
point, as shown above to the right.
on the second
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4. Measure the slope of the line
Press F6 and choose 4:Slope, move the pencil
on top of one line until the screen reads THIS
LINE, press ENTER . Label the slope with
slope b=
5. Use the two ordered pairs and the slope
formula to verify that the slope is 0.50. Show
your work below.
6. Create a parallel line
Press F4 and choose 2:Parallel Line, move the pencil on the line until it reads PARALLEL TO
THIS LINE, press ENTER , move the pencil away to where you want your new line and press
ENTER , label the new line a.
7. Measure and label the slope of line a. Drag the original line (not at a defined point) changing the tilt
(slope) of it. Compare the changes in the slopes of the two lines.
8. Make an IF...THEN...conjecture below about your findings in step 7.
Note: Try dragging the second line that was created to be parallel to the original. What is different? . . . The slope must
remain the same as the original line because it was created dependent upon it and the calculator remembers how things are
constructed. Therefore, if you wish to change things around, you must drag the original, not the dependent.
9. Use the Format tool and turn off the Coordinate Axes and Grid (to make the screen neater for
measurements).
10. Create a transversal, a line which crosses
both of the parallel lines. Measure one pair of
corresponding angles as shown in the diagram to
the right. Note: You can create points on the
lines as you measure the angles, including the
intersection points. Move the measurements
inside the angle as shown to the right. Drag line
b around and observe the change in the angle
measurements of the corresponding angles.
11. Make an IF...THEN...conjecture below about
your findings in step 10.
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12. How many more pairs of corresponding angles exist in the diagram? Without measuring, explain
below why each pair should be congruent.
EXPLORATION 3-7. PERPENDICULAR LINES.
Objective: Determine the properties which determine perpendicular lines.
1. CREATE A NEW DOCUMENT with the variable perpline.
2. Create an oblique line, labeled n.
3. Create a perpendicular line
Press F4 and choose 1:Perpendicular Line, move the pencil on top of line n, and press ENTER
twice (once to identify the line and another to identify the location).
4. Measure one of the angles created by the intersecting lines. Drag line n around and observe changes
in the angle measurement.
5. Make an IF...THEN...conjecture below about your findings in step 4.
6. Measure and label the slopes of the lines.
Using the calculate tool in F6 , calculate and use
a comment box to label the product of the slopes
as shown to the right. Drag line n around and
observe changes in the product of the slopes.
7. Make an IF...THEN...conjecture below about
your findings in step 6.
EXPLORATION 3-8. PERPENDICULAR BISECTOR.
Objective: Construct and understand the properties of a perpendicular bisector using three methods.
1. CREATE A NEW DOCUMENT with the variable perpbis.
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2. Create a perpendicular bisector
Create an oblique segment with endpoints labeled
b and s. Press F4 and choose
2:Perpendicular Bisector, move the pencil on
top of the segment until the screen reads
PERPENDICULAR BISECTOR OF THIS
SEGMENT, and press ENTER . This line
shown to the right is called the perpendicular
bisector of segment bs.
3. Create a midpoint on segment bs labeled i. Construct a perpendicular line thru point i. Drag the
endpoints of the segment around and compare this new line with the line in step 2.
4. Make an IF...THEN...conjecture below about the properties of a perpendicular bisector.
5. Clear the screen and create a small oblique segment with endpoints a and b near the center of the
screen.
6. Create two circles that have their centers and radius points at the endpoints of the segment as shown
in the diagram below to the left. Construct a line passing thru the intersections of the circles. Label the
points of intersection c and d. Create and label the point of intersection of segment ab and the line as
shown in the diagram below to the right.
7. Line CD is supposed to be the perpendicular bisector of segment AB. Make calculations on the
screen and drag the segment's endpoints to verify this statement. Explain your process and affirmation
below.
EXPLORATION 3-8P. FIND THE HIDDEN TREASURE
Objective: Use rotations and perpendicular bisectors to locate a specific spot on a map.
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1. CREATE A NEW DOCUMENT with the variable treasure.
An old map describes the location of buried treasure thus: "On the island there are only two trees, A
and B, and the remains of a ship. Start at the ship and count the steps required to walk in a straight line
to tree A. At the tree turn 90 degrees to the left and then walk forward the same number of steps. At the
point where you stop drive a spike into the ground. Now return to the ship and walk in a straight line,
counting your steps, to tree B. When you reach the tree, turn 90 degrees to the right and take the same
number of steps forward, placing another spike at the point where you stop. Dig at the point exactly
halfway between the spikes and you will find the treasure."
2. Explain how to construct a diagram that will locate the position of the treasure using rotations.
3. When you get to the island you find the ship missing! Is there any way you can still get to the
treasure? Explain your strategy. No, you may not dig up the entire island; you may dig only once!!!
4. Will the treasure be in the same position if the ship is in a different position? Will the treasure be in
the same position if the trees are in a different position? Explain why or why not.
5. Will your strategy still work if the ship is in a different position? Will your strategy still work if the
trees are in a different position? Explain why or why not.
EXPLORATION 4-1. REFLECTING POINTS.
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Objective: Understand the properties of a reflection.
1. CREATE A NEW DOCUMENT with the variable reflect.
2. Place an oblique line on the screen. Place a point labeled p close to, but not on, the line.
3. Reflect point p over the line
Press F5 and choose 4:Reflection, move the
pencil over point p until the screen reads
REFLECT THIS POINT, press ENTER ,
move the pencil over the line until the screen
reads WITH RESPECT TO THIS LINE, press
ENTER , and label the new point p' (press
2nd , 2 to access the catalog of terms and
symbols, then move up until you find the prime
symbol ', and press ENTER . It is under  and
above " ).
4. Move point p around the screen and describe the changes below. What if point p is on the line?
5. Create a line segment connecting points p and p'. How is the reflecting line related to the segment
pp'? Make calculations on the screen and drag the segment's endpoints to verify this statement.
Explain your process and affirmation below.
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EXPLORATION 4-2. REFLECTING FIGURES IN A COORDINATE PLANE.
Objective: Understand reflection in a coordinate system.
1. CREATE A NEW DOCUMENT with the variable coordin.
2. Use the Format tool to turn on the coordinate axes to a rectangular grid. Change the unit label on
the axes to equal 1 as shown below to the left.
3. Create ∆ABC with vertices on the coordinates {(-1,3),(5,2),(5,-1)} as shown on the diagram above to
the right. Press ENTER each time the screen reads ON THIS POINT OF THE GRID.
4. Reflect the triangle over the y-axis, label the new coordinates A', B', and C', and record the
coordinate of the three points below.
A
coordinates (-1,3)
A'
B
(5,2)
B'
C
(5,-1)
C'
5. If you did not have a calculator, explain in a complete sentence below how you would find the
coordinates of a figure reflected over a line.
6. Suppose the triangle is reflected over the horizontal line y=1, predict the coordinates of the vertices
of the image without using the calculator.
predicted
A
(-1,3)
A'
B
(5,2)
B'
C
(5,-1)
C'
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coordinates
7. Undo ( ◊ , z ) the image ∆A'B'C', draw the line which represents y=1, and reflect the triangle over
the line to check your answer.
EXPLORATION 4-3. REFLECTIONS AND MINIMUM DISTANCE
Objective: Use reflections to find a minimum distance along a path.
1. CREATE A NEW DOCUMENT with the variable swimmer.
2. A long pier, AE , and a shorter pier, CD ,
extend perpendicularly into the ocean with a few
miles of shore between the two piers, ED . A
swimmer wishes to swim from the end of the
longer pier (point A) to the end of the shorter pier
(point C) with one rest stop on the beach (point
B). Your goal will be to find the spot on the
beach that would create the shortest possible
swim.
Draw ED first and then construct the two perpendicular lines (use 1: Perpendicular Line -- don't
just draw segments!) Create overlapping
segments.
3. Hide the lines as shown in the diagram to the
right. Recall that when you use the Hide/Show command,
select the command from the menu, then select the object.
Also remember to hit escape after pressing enter to see the
line disappear. Find the three lenths needed ( The
scale will be 1 cm = 1mile) and adjust the points until
the measurements match those in the diagram
above.
4. Make two segments from A to a point on the
shore and from C to a point on the shore. Label
this point B. Notice in the picture that these two
segments are dotted. To do this, select F7 and
9: Dotted. Press ESC when done.
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5. Measure and label the distance of the 1st swim
(AB) and the second swim (BC). Use the
Calculate tool to find the distance of the total
swim, as shown to the right. Use 5: Comment
under F7 for the measurement descriptions.
6. Measure the distance from the rest stop on the
beach to the intersection of the beach and the pier
(point D). Drag point B along the shoreline until
you find the spot on the beach/land that will
allow
the least amount of total swimming. Leave B at this optimal spot and record the total swimming
distance and the distance from the shorter pier (along the beach).
Minimum total swimming distance (AB + BC) =
Distance from the shorter pier on the beach (BD) =
7. Now you will see the powerful use of a
reflection in this application.
Reflect point C over the beach, creating C'.
Create segment AC' as shown to the right, and
observe the results.
Note: The position of point B on your screen should be
different from the screen shown to the right.
8. Explain below how you would solve this problem without a calculator. Also explain why this
method helps you find the correct answer.
EXPLORATION 4-4.
TRANSLATIONS: COMPOSING REFLECTIONS OVER PARALLEL LINES.
Objective: Understand composition of reflections over parallel lines.
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1. CREATE A NEW DOCUMENT with the
variable compref.
2. Create a triangle to the left of two parallel lines
as shown to the right.
d1
3. Construct a perpendicular line. Create an
intersection point where the lines intersect. (One
should already be there) Find the distrance between
the two points along the perpendicular. Move the
lines so they are approximately 1.5 cm apart. We
will call this distance d1. (You don't need to label this)
4. Reflect the triangle over the line closest to the
triangle, and reflect the image over the next line
as shown in the diagram to the left.
5. Construct a line from the vertex of the original
triangle perpendicular to one of the parallel lines
as shown in the diagram above to the right.
Measure the distance between the same vertex in
the original triangle to its corresponding vertex of
the last triangle (under two reflections). This
distance is shown as d2 in the diagram to the
right.
d1
d2
Drag the original line around and modify the
distance between the reflecting lines to compare
the two measurements d1 and d2.
6. Complete the following conjecture: If a figure is reflected twice over parallel lines, then the final
image will be a t _ _ _ _ _ _ _ _ _ _, or s _ _ _ _, of _ _ _ _ _ the distance between the parallel lines.
EXPLORATION 4-5.
ROTATIONS: COMPOSING REFLECTIONS OVER INTERSECTING LINES.
Objective: Understand composition of reflections over intersecting lines.
1. CREATE A NEW DOCUMENT with the variable compref2.
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2. Create ∆PAL to the left of two intersecting lines which intersect at T as shown below to the left.
3. Reflect the triangle over the closest line. Reflect the image over the next line and label ∆P"A"L".
Hide the intermediate image and its vertices as shown in the diagram above to the right.
4. Measure and label the angle between the intersecting lines. Measure and label < ATA", the angle
between the original triangle and its second image. Drag the lines to modify the angle sizes and
compare the two measurements.
5. Complete the following conjecture: If a figure is reflected twice over intersecting lines, then the final
image will be a clockwise r _ _ _ _ _ _ _ of ____________ the angle between the intersecting lines.
6. Now you will begin confirming your conjecture. Select F7
and choose 9:Dotted, place the pencil
over the image triangle and press ENTER .
7. The calculator rotates counterclockwise, so you need to rotate by a negative angle. Use the
calculate feature and multiply < ATA" by -1. This value will be used in the next step.
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8. Rotate an object about a point with a specified angle
Press F5 and choose 2:Rotation, move the pencil over ∆PAL until the screen reads ROTATE
THIS TRIANGLE, press ENTER , move the pencil over point T until the screen reads AROUND
THIS POINT, press ENTER , move the pencil over the negative angle calculation until the screen
reads USING THIS ANGLE, and press ENTER .
The rotation should place the new image on top of the dotted image, verifying your conjecture in step
6. Not convinced yet? Undo your rotation steps by pressing ◊ and z . Notice the dotted image
reappears!
EXPLORATION 4-6. TRANSLATIONS AND VECTORS
Objective: Use vectors to translate objects on the coordinate plane.
1. CREATE A NEW DOCUMENT with the
variable vector.
2. Set up the coordinate plane as in
EXPLORATION 4-2.
3. Construct ∆CHI with vertices
{(-6,-1),(-5,-3),(-4,-3)} and a vector (press F2
and choose 7:Vector) with initial point (0,0) and
terminal point (4,2) as shown to the right.
4. Translate the triangle by the marked vector
Press F5 and choose 1:Translation, move the pencil on top of the triangle until the screen reads
TRANSLATE THIS TRIANGLE, press ENTER , move the pencil on top of the vector until the
screen reads BY THIS VECTOR, and press ENTER . Label the vertices of ∆C'H'I'.
5. Move the terminal point of the vector around and observe changes in the image under the
translation. Use a complete sentence to explain below how a vector with initial point (0,0) and
terminal point (x,y) translates the triangle.
6. There are an infinite number of vectors that will translate ∆CHI. The initial point can start
anywhere, not necessarily at the origin. Suppose the vertices of ∆CHI are translated to {(-7,2),(-6,0),(5,0)} under a translation. Find the initial and terminal points of three vectors which make this
possible. Record your points below.
vector 1
vector 2
vector 3
initial point
terminal point
State in terms of # of units right/left and # of units up/down the translation of the triangle.
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How do the coordinates for your vectors above demonstrate this? Be specific as to x- and ycoordinates.
7. Astro states that he used a vector to translate the vertices of ∆CHI to {(-2,-3),(-1,-5),(1,-4)}. Can
you find a vector that Astro could have used to make this translation? If so, what is it? If not, explain
why not.
EXPLORATION 4-7. GLIDE REFLECTIONS
Objective: Use translations and reflections to create footsteps.
1. CREATE A NEW DOCUMENT with the variable glideref.
2 Draw an oblique line thru the center of the screen and a small dinosaur footprint (3 toes) above the
line as shown below to the left. Use F3 and choose 4:Polygon to create the foot, and hide the
vertices.
3. Create a small vector that moves in the direction of the line. Reflect the foot over the line and
translate the image by the vector as shown in the diagram above to the right.
4. Create a macro to create the glide reflection
A macro allows you to perform multiple steps at once. This macro will perform the glide reflection,
the reflection and translation without showing the intermittent image. Press F4 and choose 6:Macro
Construction » 2:Initial Objects, select the first footprint (polygon), select the vector, and select the
line in the specified order until they are all dotted. Press F4 and choose 6:Macro Construction »
3:Final Objects, and select the last footprint until it is dotted. Press F4 and choose 6:Macro
Construction » 4:Define Macro, save the file name and variable as steps in your folder.
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Now execute the macro by pressing F4 and
choosing 6:Macro Construction » 1:Execute
Macro. Press ENTER when the name steps
appears on the screen, select the footprint, select
the vector, and select the line in the specified
order. A new footprint under the glide reflection
should appear on the screen after a short while.
Repeat the procedure until a series of footprints
cover the screen as shown to the right.
EXPLORATION 5-2. CONGRUENCE AND EQUALITY
Objective: Compare measurements of a pre-image and its image under an isometry.
1. CREATE A NEW DOCUMENT with the
variable congru.
2. Create a four-sided polygon MATH (using the
F3 menu) and a point on the screen as shown
to the right. Measure and label each of the angles
and sides of the polygon.
3. Your teacher will instruct you to perform one
of the following isometries: translation, rotation,
reflection, or glide reflection.
You may need to move the polygon MATH
higher on the screen so its image is completely
visible.
4. Label the image polygon M'A'T'H'. Measure the angles and segments of the image. Drag around
any of the vertices of the original figure MATH and observe the similarities after changes in
measurements between the two figures.
5. Make an IF...THEN... conjecture below based on your findings in step four.
EXPLORATION 5-4. ALTERNATE INTERIOR ANGLES
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Objective: Find the relationship between alternate interior angles in a diagram.
1. CREATE A NEW DOCUMENT with the
variable altint.
2. Create a triangle. Construct a line parallel to
one side of the triangle, passing through the
opposite vertex as shown in the diagram to the
right. Measure one pair of alternate interior
angles.
3. Drag one vertex of the triangle around so the
angle measurements change. Observe any
relationships between the alternate interior
angles.
4. Measure the second pair of alternate interior angles. Drag around a vertex of the triangle and notice
any similarities or differences from your result in step 3.
5. Make an IF...THEN... conjecture below based on your findings in steps 3 and 4.
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EXPLORATION 5-5. PERPENDICULAR BISECTOR THEOREM
Objective: Find the relationship between a segment and any point on its perpendicular bisector
1. CREATE A NEW DOCUMENT with the variable ptperp.
2. Construct a segment, its perpendicular bisector, and two points on the perpendicular bisector (one
above one below the segment) as shown in the diagram below to the left.
3. Measure the distance from the points on the perpendicular bisector to the endpoints of the segment
as shown above in the diagram to the right. You do not need to create the dotted segments; they are
shown to help you visualize the distances.
4. Drag the points along the perpendicular bisector and observe changes in the measurements.
5. Write an IF...THEN... statement below describing your findings in step 4.
EXPLORATION 5-5P. CAPTURE THE FLAG
Objective: Determine how perpendicular bisectors can be used to create a fair game.
1. CREATE A NEW DOCUMENT with the variable capture.
Three teams, A, B, and C, each start from a location on a field. Their goal is to grab the flag located in
a
known location on the field. If the game is fair, then each team has to run the same distance to the flag.
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2. Where should the flag be positioned in a fair game? Describe how you found the position. Explain
why this method works.
3. Will your solution change if the field were in a different shape, like a rectangle, circle, or blob?
Explain why or why not.
EXPLORATION 5-7. SUM OF ANGLES IN A POLYGON
Objective: Find the relationship between the number of sides in a polygon and the sum of its angles
1. CREATE A NEW DOCUMENT with the variable polysum.
2. Create a triangle, measure all of its angles, calculate their sum and record it below. Drag around a
vertex to confirm the angle sum remains constant.
3. Clear the screen and repeat this procedure for polygons up to six sides and record the information
below.
name of polygon
triangle
quadrilateral
number of sides
sum of angles
3
4
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pentagon
hexagon
5
6
4. There are two types of formulas which will predict the sum of the measures of the angles in a
polygon. A recursive formula uses the constant difference in the sum of angles for every increase in
one side of the polygon.
Fill in the blank:
Sum of angles in polygon = Sum of angles in polygon + ________˚
with n-sides
with (n-1) sides
Use the recursive formula to determine the sum of the angles in a heptagon, a seven-sided polygon.
Show your work below.
5. An explicit formula predicts the sum of the
angles in any polygon, even if you do not know
the previous polygon angle sum. Let n represent
the number of sides in any polygon. Use the
diagram at the right to think about the value (in
terms of n) that represents the number of triangles
that share vertices within any polygon. Also, use
the table you developed in step 3 to help test your
explicit formula.
Fill in the blanks:
Sum of angles in polygon = ______ * (
)
with n-sides
number of
triangles
Use the explicit formula to confirm the sum of the angles in a heptagon that you found in step four.
Show your work below.
6. Clear the screen, create a heptagon, measure its angles, and check your answers to questions 4 and 5.
EXPLORATION 6-1. REFLECTION-SYMMETRIC FIGURES
Objective: Determine properties of reflection-symmetric figures.
1. CREATE A NEW DOCUMENT with the variable refsym.
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2. Create a vertical line on the screen. Draw the
left hand side of a face with polygons, circles, and
arcs as shown to the right.
Create an arc
Press F3 and choose 2:Arc, move the pencil to
a starting position, press ENTER , move the
pencil a little more, press ENTER , and move
the pencil one more time until the desired arc, and
press ENTER . Move the endpoints of the arc
if
you would like to adjust its length. Move the middle point of the arc if you would like to adjust its
curvature.
3. Reflect each of the objects over the line until
your face is complete.
4. Adjust the size and shape of the ears, eyes,
nose, and lips on the LEFT SIDE of the screen
and observe the changes on the right side. For
smoother movement, you may want to move the
pointer on top of a point, press
2nd , HAND KEY to LOCK, and drag
using the keypad. Press ESC to return to the
pointer when you are finished dragging.
5. The vertical line is called a line of symmetry. The face is reflection-symmetric about this line.
State at least two properties of a figure that is reflection-symmetric (without using reflection or
symmetric in your explanation).
6. Drag the symmetry line until it is oblique and
the face looks distorted. Construct a segment
with the middle of the eyes as endpoints. What
geometric property will the symmetry line have
on this segment? Verify this conjecture by
making appropriate measurements and/or
calculations. Describe your findings below.
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EXPLORATION 6-2. ISOSCELES AND EQUILATERAL TRIANGLES
Objective: Determine properties of isosceles and equilateral triangles.
1. CREATE A NEW DOCUMENT with the variable isostri.
2. Create scalene ∆ABC (with unequal side lengths), measure and label segments AB and AC as shown
below to the left.
3. Construct the midpoint D of side BC. Construct the median of BC (the segment connecting A and
D), the perpendicular bisector of BC, and the angle bisector of < CAB as shown in the diagram above
to the right.
4. Drag the vertex A until the median, perpendicular bisector, and angle bisector coincide. What type
of triangle results? Explain below how you determined this result.
5. Measure the base angles, < ACB and < ABC. Write an IF...THEN... statement below describing the
condition in step four and your results in step five.
6. Create the perpendicular bisectors of the other sides of the triangle, segments AC and AB. Drag the
vertices of the triangle until the triangle is reflection symmetric with respect to all of the perpendicular
bisectors.
7. Measure the remaining side and angle in the triangle. What type of triangle results? Explain below
how you determined this result.
EXPLORATION 6-2P. SHARK ATTACK.
Objective: Determine the ideal position within an equilateral triangle given a series of conditions.
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1. CREATE A NEW DOCUMENT with the variable shark.
Guppy, Tadpole, and Goldfish beach surround Equilateral Sea, a perfect equilateral triangle. Tiger
shark and his family swim together in the Equilateral Sea. When the family gets hungry, Tiger makes
sure they stay in one place (so he doesn't lose them), and then finds food at each of the beaches. Tiger
gets food at one beach at a time because he can only hold so much food in his mouth at a time. So he
will get food at one beach, come back to share the food (small square inside the triangle), and then
repeat the process at the other beaches.
2. Use rotations to construct Equilateral Sea, and place a point inside the triangle to represent the
hungry sharks. Construct perpendicular lines from the sharks to each of the sides of the triangle and the
intersection points on the beaches.
3. Where should Tiger place his family in the Equilateral Sea in order to swim the least distance for his
three hunting trips? Explain your reasoning.
4. What is the sharks lived in Isosceles Sea? Would your results from step 3 change? Why or why not?
Explain your analysis.
EXPLORATION 6-3. CONSTRUCTING PARALLELOGRAMS
Objective: Construct a parallelogram and rectangle. Modify the parallelogram into a rhombus and the
rectangle into a square.
1. CREATE A NEW DOCUMENT with the variable pararect.
2. Create an oblique line and a parallel line approximately 3 cm away. Connect the points on each line
with a segment as shown in the diagram below to the left.
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3. Construct a point on the line to the left. Construct a line parallel to the segment passing through this
point, and create a point of intersection as shown in the diagram above to the right.
4. Label the vertices M, A, T, and H. Construct
overlapping segments and hide the three lines
until your parallelogram matches the one to the
right. It's quickest to hold down the shift key, select all
three lines, and then choose the hide command. Remember
to his ESC after you hide the lines.
5. Measure the slopes of the sides of the figure.
Drag the vertices around until the slopes change.
Is the figure still a parallelogram? Explain how
you know.
6. Delete the slope measurements. Measure the length of each side of MATH. Drag the vertices until
each of the sides have the same length. This special type of parallelogram is called a rhombus.
Once you drag the vertices again, the parallelogram is no longer a rhombus because the sides will not
have the same length. Your goal in chapter seven of the text is to construct a rhombus once you
familiarize yourself with its properties.
7. Clear the screen. Create an oblique line and then construct a perpendicular with a point of
intersection C. Place point U on one line and point S on the other line as shown below to the left.
8. Draw perpendicular lines passing thru points S and U. Label the point of intersection B as shown in
the diagram above to the right.
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9. Create overlapping segments CU, UB, BS, and SC. Hide the four lines. Choose Hide/Show, and then
select the four lines by pressing ENTER.
10. Measure the length of each side of the figure CUBS. Drag points U and S until the measurements
of all of the sides are equal. The most specific name for this figure is a square because all of its angles
and sides are equal. This quadrilateral can be named a parallelogram, a rectangle, and a rhombus.
Make necessary measurements on the calculator to justify why these names are also applicable to the
figure. Explain below.
11. Once you drag the vertices again, the rectangle is no longer a square because the sides will not
have the same length. Clear the screen and use rotations to construct a square that will always remain
a square even after you drag around the endpoints. Explain the steps to your construction below.
Recall that you must create a numerical edit (under F7 ) for the number of degrees you wish to rotate the object.
EXPLORATION 6-4. CONSTRUCTING A KITE
Objective: Construct a kite using three different methods.
1. CREATE A NEW DOCUMENT with the variable kite.
2. Construct a kite using two intersecting circles as shown in Figure I of your text on page 323. Hide
the circles and leave only the kite. Make necessary measurements on the calculator to justify why this
is a kite. Explain below.
3. Clear the screen. Construct a kite using a reflected triangle over a symmetry line as shown in
Figure II of your text on page 323. Make necessary measurements on the calculator to justify why this
is a kite. Explain below.
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4. Modify the vertices of the kite to make it
convex (if necessary). Create the second diagonal
of the kite and label the intersection of the
diagonals F as shown in the diagram to the right.
5. Explain in a complete sentence below how the
diagonals of a kite relate to each other. Make
necessary measurements on the calculator to
justify your answer below. Differentiate the
diagonals by calling CD the symmetrical
diagonal.
6. Clear the screen. Construct another kite using a different method based on your explanation in step
5. Use an exploration from chapter five for additional assistance. Explain the steps to your
construction below.
EXPLORATION 6-5. CONSTRUCTING A TRAPEZOID
Objective: Construct a trapezoid and an isosceles trapezoid, and explore their properties.
1. CREATE A NEW DOCUMENT with the variable trapez.
2. A trapezoid is a quadrilateral with at least one pair of parallel sides. Construct a trapezoid that is not
a parallelogram. Explain the steps to your construction below. Make necessary measurements on the
calculator to justify why this is a trapezoid and explain below.
3. Measure and label a pair of consecutive angles
between a pair of parallel sides, as shown in the
diagram to the right.
4. Drag the endpoints of the original segment
until you see the angles change measurement.
What relationship exists between these angles?
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5. Clear the screen. An isosceles trapezoid is a trapezoid that has a pair of base angles equal in
measure. Construct an isosceles trapezoid using the Isosceles Trapezoid Symmetry Theorem and the
drawing related to the theorem on page 330 of your text. Hint: Use a Reflection. Explain the steps to
your construction below.
6. Measure the length of the legs, the non-parallel sides, of the isosceles trapezoid. Drag around the
vertices until the measurements change. What relationship exists between the legs of an isosceles
trapezoid?
EXPLORATION 6-M. MYSTERY QUADRILATERALS
Objective: Determine the most specific name for each mystery quadrilateral (parallelogram, rectangle,
square, rhombus, trapezoid, isosceles trapezoid, kite, or quadrilateral).
You must provide a valid “proof” for
each mystery quadrilateral. For each
quadrilateral, write a short paragraph
(see below for an example). Support
your answer with the measurements you
have taken. Make sure that you list all
relevant measurements on your lab. Also
be sure to state why the quadrilateral
could NOT be any other name.
Quadrilateral
Before getting started, fill in your
hierarchy to refer back to if needed!
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Example Solution Paragraph:
m BA = 3.06 cm
Quadrilateral ABCD is a kite because
A
consecutive sides are congruent.
Opposite sides are NOT parallel because
their slopes are different. Even though m AD =
there is a right angle, ABCD can not be a
rectangle if it is not a parallelogram.
Slo pe AD = -3.54
B
m BC = 3.06 cm
2.40 cm
Slo pe BC = -12 .81
Slo pe DC= 0.28
Slo pe BA = 0.51
m ADC = 90 .00°
C
D
m DC= 2.40 cm
1. Open QUAD1 in the MYSTERY folder.
MYSTERY QUADRILATERAL #1
Solution Paragraph.
sketch of diagram and relevant measurements!
2. Open QUAD2 in the MYSTERY folder.
MYSTERY QUADRILATERAL #2
Solution Paragraph.
sketch of diagram and relevant measurements!
3. Open QUAD3 in the MYSTERY folder.
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MYSTERY QUADRILATERAL #3
Solution Paragraph.
sketch of diagram and relevant measurements!
4. Open QUAD4 in the MYSTERY folder.
MYSTERY QUADRILATERAL #4
Solution Paragraph.
sketch of diagram and relevant measurements!
5. Open QUAD5 in the MYSTERY folder.
MYSTERY QUADRILATERAL #5
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Solution Paragraph.
sketch of diagram and relevant measurements!
6. Open QUAD6 in the MYSTERY folder.
MYSTERY QUADRILATERAL #6
Solution Paragraph.
sketch of diagram and relevant measurements!
EXPLORATION 6-6. ROTATION SYMMETRY
Objective: Compare the number of vertices in an object to its angle of rotation.
1. CREATE A NEW DOCUMENT with the variable rotsym.
2. Place point A in the middle of the screen.
Create an arc, starting at point A, and ending near
the top of the screen. Create and measure an
angle in the lower left-hand side of the screen as
shown in the picture to the right.
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3. Rotate the arc about point A using the angle.
Repeat the arc rotations until the arc has gone
completely around (make seven arcs). Note that
the arcs may not perfectly overlap.
4. Create a seven sided polygon (heptagon) which
has vertices touching the ends of each arc as
shown in the diagram to the right.
5. Adjust the angle in the lower-left hand side of
the screen until there is exactly six
spokes/vertices as shown above in the picture to
the right (two arcs/vertices overlap). The arcs do
not have to be perfect, but should be close to
overlapping. Record the information on the next
page.
#arcs/vertices
6
5
4
name of polygon
hexagon
pentagon
quadrilateral
degree of rotation
3
triangle
6. Adjust the angle in the lower-left hand side of the screen until there are exactly five spokes/vertices.
Record the information above. Repeat the procedure until the table above is complete.
7. The overlapping arcs/vertices occur in this exploration because you are creating a special type of
polygon. Make up a name that you would use to classify these polygons.
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Make necessary measurements on the calculator to justify why you have chosen this name. Explain
below.
8. Write an equation below relating the number of vertices (V) in this special type of polygon
compared to the degree rotation (D) between consecutive vertices in the polygon. Explain how you
determined this.
EXPLORATION 6-7. REGULAR POLYGONS
Objective: Determine properties of a regular polygon
1. CREATE A NEW DOCUMENT with the variable regpoly.
2. Create a regular pentagon that will cover a
large portion of the screen as shown to the right.
Press F3 and choose 5: Regular Polygon,
move the pencil to the center of the screen, press
ENTER to locate the center, move out from
this center (you'll see a dotted circle) as far as you
can before part of the circle goes off the screen.
Press ENTER . Move the pencil away from the
point on the dotted circle until the screen reads
{5} and press ENTER . You will see a regular
pentagon as shown to the right.
3. Measure the side lengths and angles of the regular pentagon. Explain below two properties of
regular polygons which you discovered.
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4. Connect seven segments from the center of the
pentagon to each of its vertices. Five triangles
should form as shown in the diagram to the right.
Do you know which company uses this design as
their logo?
5. Without measuring on your calculator,
determine the measure of each of the three angles
in one of these triangles. Explain your reasoning
and show your work below. Hint: Look back at
Exploration 6-6.
6. Check your answer to step five by measuring each angle of one of the triangles. If your answers do
not match, go back and check your work.
EXPLORATION 7-2. CONGRUENT TRIANGLES
Objective: Determine arrangements to determine congruent triangles
1. CREATE A NEW DOCUMENT with the
variable congtri.
2. Create two different looking triangles TRI and
AGN as shown to the right.
3. Measure the lengths of the three sides in both
triangles. Move the vertices of ∆TRI until
TIAN, TRAG, and RIGN. It's OK if you are
within 0.01 cm away. Is this the only possible
arrangement for the triangle to satisfy this
condition (excluding rotations)?
Two figures are  if their corresponding sides
AND angles are . You have satisfied the
condition that the corresponding sides are ; now
you must verify that the corresponding angles are
also .
4. Measure and compare the corresponding angles
of the two triangles. Given that three
corresponding sides are congruent (SSS), is this
enough to verify that ∆TRI∆AGN? Compare
your results with a classmate.
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5. Clear the screen. Recreate triangles TRI and
AGN as shown in step two. Measure the angles
in ∆TRI until <RTI<NAG, <RIT<NGA, and
<TRI<ANG. It's OK if you are within 0.15˚. Is
this the only possible arrangement for the triangle
to satisfy this condition (excluding rotations)?
6. Measure and compare the corresponding sides
of the two triangles. Given that three
corresponding angles are congruent (AAA),
is this enough to verify that ∆TRI∆ANG? Compare your results with a classmate.
7. Clear the screen. Recreate triangles TRI and
AGN as shown in step two. Measure the angles
and sides in ∆TRI until <RTI<NAG, TRAN,
and <TRI<ANG. It's OK if you are within 0.01
cm or 0.15 degrees. Is this the only possible
arrangement for the triangle to satisfy this
condition (excluding rotations)?
8. Measure and compare the remaining corresponding sides and angles of the two triangles. Given that
two corresponding angles and an included side are congruent (ASA), is this enough to verify that
∆TRI∆ANG? Compare your results with a classmate.
9. Clear the screen. Recreate triangles TRI and
AGN as shown in step two. Measure the sides
and angles in ∆TRI until TIAN, <TIRANG,
and RIGN. It's OK if you are within 0.01 cm or
0.15 degrees. Is this the only possible
arrangement for the triangle to satisfy this
condition (excluding rotations)?
10. Measure and compare the remaining corresponding sides and angles of the two triangles. Given
that two corresponding sides and an included angle are congruent (SAS), is this enough to verify that
∆TRI∆AGN? Compare your results with a classmate.
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11. Clear the screen. Recreate triangles TRI and
AGN as shown in step two. Measure the angles
and sides in ∆TRI until <RTI<NAG,
<TRI<ANG, and RING. It's OK if you are
within 0.01 cm or 0.15 degrees. Is this the only
possible arrangement for the triangle to satisfy
this condition (excluding rotations)?
12. Measure and compare the remaining corresponding sides and angles of the two triangles. Given
that two corresponding angles and a non-included side are congruent (AAS), is this enough to verify
that ∆TRI∆ANG? Compare your results with a classmate.
EXPLORATION 7-5. SSA CONDITION AND HL CONGRUENCE
Objective: Examine triangles in the SSA condition and verify the condition for HL congruence.
1. CREATE A NEW DOCUMENT with the variable ssahl.
2. Create a circle with center A, and radii AB and AC, as shown on the next page.
3. Create a line passing thru points B and C. Create segment AD from A to a point on the line, left of
point B, as shown in the diagram to the right of the previous page
.
4. Create overlapping segment CD. Hide the line
and make the circle dotted as shown in the
diagram to the right.
5. ∆ABD and ∆ACD are overlapping. Mentally
separate and redraw the triangles below so they
are not overlapping.
6. Place tick marks on the triangles to identify pairs of congruent angles and congruent sides (you
should have three). Justify below why each pair of corresponding parts is congruent.
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7. Given that two corresponding sides and a non-included angle are congruent (SSA), is this enough to
verify that ∆DBA∆DCA? Explain.
8. ∆DBAand ∆DCA will be congruent under one situation. Drag point B around the circle until the
two triangles are congruent. When ∆DBA∆DCA, what type of triangles are on the screen? Explain
your reasoning below.
9. Re-evaluate your explanation in step 7. Explain below why this might be called HL congruence
instead.
10. Wait! There's even another case. Construct a
Star Trek symbol inside a circle as shown to the
right. Modify point S until <AST=<ASR (or
within 0.15˚).
11. Place tick marks on the triangles to identify
pairs of congruent angles and congruent sides
without any additional measuring (you should
have three). Justify below why each pair of
corresponding parts is congruent.
12. Measure and compare the remaining corresponding sides and angles of the two triangles. Given
that two corresponding sides and a non-included angle are congruent (SsA), is this enough to verify
that ∆DBA∆DCA? Explain.
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This case in step 12 worked and the case in step seven did not work because the sides opposite the
congruent angles in each triangle were longer than the other pair of congruent sides. Therefore, SsA
will be used to distinguish step 12 from step 7 (SSA).
EXPLORATION 7-6. TESSELLATIONS
Objective: Construct tessellations with triangles using rotations and translations.
1. CREATE A NEW DOCUMENT with the variable tessell.
2. Follow along with Activity 1 on page 398 of your text. Create ∆ABC near the lower left side of
your screen. Create the midpoint M on side AC as shown below to the left.
3. Create a Numerical Edit for rotation purposes
Press F7 and choose 6:Numerical Edit. Move the pencil near the upper-right hand portion of the
screen, press ENTER , type 180, press ESC . This will be used for rotating the ∆ABC by 180˚.
4. Rotate ∆ABC about point M by 180˚, as shown above in the diagram to the right.
5. Create overlapping vector BC with initial point B and terminal point C. Press F2 and choose
7: Vector. Select point B first, then point C. This will allow you to move the figure to the right.
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6. Translate ∆ABC by vector BC. Press F5 and
choose 1: Translation. Select the triangle and
then the vector. Then translate the rotated
triangle by vector BC. Continue the process until
you have created eight triangles total, as shown to
the right. Drag around the vertices of the triangle
to see if the triangles remain connected.
7. Create overlapping vector BA with initial point
B and terminal point A. This will allow you to
move the figure up.
8. Translate ∆ABC by vector AB. Translate the
rotated triangle by vector AB. Continue the
process until you have created two rows with
sixteen triangles total, as shown to the right.
Drag around the vertices of the triangle to see if
the triangles remain connected.
9. Which kind of special quadrilateral is ABCD?
How do you know?
10. Clear the screen. Try to build the tessellation
to the right. Use Activity 2 on page 399 of your
text as a guideline for hints. Drag around the
vertices of ∆XYZ to see if the tessellation
remains connected.
EXPLORATION 7-7. PROPERTIES OF PARALLELOGRAMS
Objective: Discover properties of parallelograms.
1. CREATE A NEW DOCUMENT with the variable pgram.
2. Construct parallelogram CUBS. Make sure the figure remains a parallelogram after you drag around
its endpoints. Don't worry about doing overlapping segments and hiding the lines. Just create intersection points.
Describe how you constructed your parallelogram.
3. Measure the lengths of the two pairs of opposite sides. Drag around one of the vertices until you see
the measurements change. Write a property of parallelograms below based on your observations.
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4. Delete the length measurements. Measure the two pairs of opposite angles. Drag around one of the
vertices until you see the measurements change. Write another property of parallelograms below based
on your observations.
5. Delete the angle measurements. Create the diagonals of the parallelogram. Make necessary
measurements on the calculator to determine a property of the parallelogram's diagonals. Hint: Create a
point at intersection between the two diagonals and investigate the lengths of each part of the diagonals. Drag around
one of the vertices to make sure the property is true in general.
EXPLORATION 7-7P. BOUNCING OFF THE WALLS
Objective: Determine the relationship between parallel lines and a position on the triangle.
1. CREATE A NEW DOCUMENT with the variable bounce.
Jeff is in the triangular room shown to the right.
He walks from a point on AC parallel to BC.
When he reaches AB, he turns and walks parallel
to AC. When he reaches BC, he turns and walks
parallel to AB. He continues this process until he
returns to his starting point.
2. Construct a diagram to represent this situation.
Make sure the lines remain parallel to the sides,
even after you drag around the vertices of the
triangle or change Jeff's starting point. Create
segments to overlap the parallel lines and hide the
lines that fall outside of the triangle.
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3. What is the smallest number of walls that Jeff will touch before he reaches his starting point? What
is his starting point in this situation? Why will this specific starting point result in the least number of
bounces?
4. What is the largest number of walls that Jeff will touch before he reaches his starting point? Will
Jeff always wind up in his original starting position? Explain why or why not.
EXPLORATION 7-9. EXTERIOR ANGLES
Objective: Compare angles and sides in a triangle, and explore the relationship between interior and
exterior angles.
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1. CREATE A NEW DOCUMENT with the
variable extang.
2. Construct ∆CAT on top of an oblique line with
exterior point S, as shown in the diagram to the
right. Measure the sides and angles of ∆CAT.
3. Fill in the table below comparing each side of
the triangle with its opposite angle. Use YOUR
measurements.
Side Length
Opposite Angle Measurement
AT=
ACT=
.
AC=
ATC=
.
CT=
CAT=
.
Rank the three side lengths 1,2, and 3 from longest to shortest. Next, rank the angle measurements
opposite those sides 1,2, and 3 from largest to smallest. What do you observe?
Now test this hypothesis by dragging around a vertex and seeing if it remains true. Does it?
4. Measure exterior angle ATS. Describe below how this angle compares with the measurements of
alternate interior TAC and TCA. Drag around point T to see if this is still true.
5. Compare the exterior angle with the sum of its alternate interior angles.
Exterior Angle Measurement
Sum of Interior Angles at other two vertices
ATS =
ACT + CAT =
Drag around the vertices of the triangle and try it again. Is the outcome the same? Explain below why
or why not this outcome would be the same. Hint: Compare the sum of the angles in a triangle to a
straight angle.
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EXPLORATION 8-1. PERIMETER OF A REGULAR POLYGON
Objective: Determine the perimeter of a regular polygon with n sides.
1. CREATE A NEW DOCUMENT with the variable regperim.
2. Use the regular polygon tool to create an equilateral triangle. Measure the perimeter and length of
one side (distance between consecutive vertices), and record the information in the table below.
Name of regular polygon
number of sides
length of one side
perimeter
equilateral triangle
3
__________
__________
square
4
__________
__________
hexagon
__________
__________
octagon
__________
__________
3. Repeat the procedure in step 2 with a square, regular hexagon, and regular octagon. Record the
information in the table above.
4. Is there a pattern in the values in the table above? Explain below.
5. If the length of one side of a regular n-gon is represented by s, state its perimeter p in terms of n and
s in the form of an equation.
EXPLORATION 8-2. AREA OF A RECTANGLE
Objective: Determine the area of a rectangle given a length and width.
1. CREATE A NEW DOCUMENT with the
variable rectarea.
2. Construct a rectangle without hiding any of the
lines. Use the polygon tool to create an
overlapping rectangle so the vertices of the
polygon meet the vertices of the rectangle that
you are creating. The polygon tool allows you to
measure area.
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3. Hide the lines which created the rectangle.
Only the polygon should be on the screen. Drag
around the vertices of the polygon to ensure the
figure remains a rectangle.
4. Measure the lengths of two consecutive sides,
and the area, of the polygon as shown in the
figure at the right.
5. Use either addition, subtraction, multiplication,
or division, with the calculate tool to determine a
relationship between the length, width, and area
of the rectangle. Write this relationship below in
a formula.
AREA =
EXPLORATION 8-2P. OPTIMAL QUADRILATERALS
Objective: Determine the quadrilateral with fixed perimeter that has maximum area.
1. CREATE A NEW DOCUMENT with the variable optimal.
2. Create a quadrilateral using the polygon tool.
3. Measure the perimeter of the quadrilateral and drag the vertices until the perimeter is 8.0 cm.
4. Measure the area of the quadrilateral, and record the value in the table below. Drag the vertices of
the quadrilateral until it changes shape, but still has a perimeter of 8.0 cm. Try various types of
quadrilaterals and record the new area below. Continue this process for a variety of different
quadrilaterals.
Quadrilateral
Area (perimeter = 8 cm)
Quadrilateral
__________
Rectangle
__________
Parallelogram
__________
Square
__________
Kite
__________
____________
__________
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5. What type of quadrilateral with perimeter 8.0 cm will have a maximum area? Justify your response.
6. Repeat the exploration using a different fixed perimeter, how do your results change? Explain your
analysis.
7. CREATE A NEW DOCUMENT with the variable optimal2.
8. Suppose the exploration is reversed so that the area of a quadrilateral is fixed to a specific number,
like 8.0 cm2. What type of quadrilateral with fixed area will have a minimum perimeter? Explain your
constructions and your analysis.
EXPLORATION 8-3. AREAS OF IRREGULAR POLYGONS
Objective: Determine the area of an irregular polygon on a coordinate plane.
1. CREATE A NEW DOCUMENT with the variable irrarea.
2. Use the Format tool to turn on the coordinate axes to a rectangular grid. Change the unit label on
the axes to equal 1 as shown below to the left.
3. Create a large W on the screen using the coordinates shown above to the right. Make sure that each
vertex falls on a grid point.
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4. Determine the area of each of the pieces by
exact calculation or estimation. Keep in mind
that the horizontal and vertical distance between
two consecutive vertices is one unit. Do not
redraw these figures on your calculator - just
examine the figures to the right.
Area of A=
Area of B=
Area of C=
Area of D=
5. Now use this method to find the area of the
letter W. For instance, the region identified in the
picture to the right might be about 0.7 cm2. Use
small pieces and the coordinates on the diagram
to the right to estimate the area of the W. Show
your work below.
6. Change the unit label on the axes to equal 0.5 (otherwise, the area will not be measured accurately).
Use the Area tool to find the exact area of the W. Check your results with your work in step 5. If your
answer is not within 0.7 cm2, then go back and check your work from step 5.
A more precise approach to finding the area of the W will be explored in section 8-5.
EXPLORATION 8-4. AREA OF A TRIANGLE
Objective: Determine the area of a triangle and relationships between the altitudes of a triangle.
1. CREATE A NEW DOCUMENT with the variable triarea.
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2. Use the Format tool to change the coordinate
axes to a rectangular grid and turn the grid ON.
Change the unit label on the axes to 1. Create
∆ABC with vertices shown to the right.
3. Use the estimation methods in section 8-3 to
determine the area of this triangle. Mark up the
diagram to the right and show your work below.
4. Change the unit label on the axes to equal 0.5.
You may need to move the origin a little lower to
see the entire triangle. Use the Area tool to
check your answer. Check your results with your
work in step 3.
5. The triangle has three bases and three heights
(altitudes). You can find the area of a triangle
with any one pair of base and height. Choose one
side of the triangle, and call it the base (AB=3
cm). The height of the triangle with base AB will
be the perpendicular distance from AB to the
opposite vertex C (height = 2 cm).
If the position of point C changes along the yaxis, the height (altitude) of the triangle always
remains the same, as shown in the diagrams to the
right.
Use the formula of a rectangle and the picture to
the lower right to determine the formula for the
area of a triangle. Write the formula below and
explain how you derived it.
You will check this formula by creating a
different altitude and measuring its corresponding
base.
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5. Create an altitude from B to AC
Create a line perpendicular to AC and passing
through B. Call the point of intersection D.
Create an overlapping segment BD, and hide the
line.
6. Measure and label the lengths of base AC and
height BD.
7. Use the Calculate tool to check your area
formula in step four with base AC and altitude
BD.
8. Construct the remaining altitudes EC and AF
as shown to the right. Hide the perpendicular
lines that you used to construct the altitudes.
9. Drag the vertices of ∆ABC around and
describe below the relationship among the
altitudes.
10. The altitudes intersect at a point called the
orthocenter. The orthocenter lies inside an acute
triangle, as shown in the example above.
Sometimes, the altitudes lie on, or outside, the
triangle. In the picture at the right, altitude BD is
inside the triangle, but the other two altitudes and
the orthocenter lie outside of the triangle.
You must unhide, or show, the perpendicular
lines in order to see an altitude outside of the
triangle. Press F7 and choose 1:Hide/Show,
move the pointer on top of each dotted
perpendicular line and press ENTER when the
screen reads THIS LINE. You should now have
three perpendicular lines showing.
11. Drag the vertices of ∆ABC around and observe changes in the location of the altitudes and
orthocenter. Name the type of triangle that has two altitudes outside of the triangle and the orthocenter
lying outside of the triangle.
Name the type of triangle that has two altitudes on the triangle and the orthocenter lying on of the
triangle.
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EXPLORATION 8-4P. TRIANGLE IN A RECTANGLE
Objective: Determine the ratio of the areas between a special rectangle and triangle.
1. CREATE A NEW DOCUMENT with the variable trirec.
2. Construct rectangle ABCD with length to
width in the ratio of 3:2. As you drag the vertices,
the ratio should always remain 3:2. (Hint: use
reflections). Explain how you created this
construction.
3. Create angle bisectors AN and BM of angle A
and angle B, respectively. Make E the point of
intersection of the angle bisectors. You should
end up with a small triangle EMN at the bottom
of the rectangle, as shown to the right.
4. What is the ratio of the area of the area of rectangle ABCD to the area of triangle EMN? Is this
always true, even when you change the size of the rectangle? Explain your analysis.
5. Let segment length MN equal 2x. Divide the rectangle in smaller sections and prove your results
from step 4. Show your work.
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6. How would the results from this investigation differ if ABCD were a parallelogram? Explain your
analysis.
EXPLORATION 8-5. AREA OF TRAPEZOIDS
Objective: Determine the area of a trapezoid, parallelogram, and an irregular polygon.
1. CREATE A NEW DOCUMENT with the variable traparea.
2. Construct a trapezoid (start with segment OI and
construct a parallel line ZD) and with altitudes passing
thru D and O, as shown in the diagram to the
right.
3. Create segment DO. Make overlapping
segments for each of the altitudes and for
segment ZD and hide any lines which were used
in the construction of the trapezoid. Drag around
the vertices to ensure the trapezoid remains intact.
Use F7 and 8: Thick for the four segments
which make up the sides of the trapezoid.
4. The altitudes passing thru O and D should be equal in length. Explain why below without
measuring.
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5. Use the Comment feature so the altitudes equal
h, and the bases are b1 and b2, as shown on the
diagram.
The area of trapezoid ZOID is the same as the
sum of which two triangles?
Predict the area of the trapezoid below in terms of
h, b1, and b2, by using the formula for area of a
triangle and your answer to the previous question.
AREA OF TRAPEZOID =
6. Use the polygon tool ( F3 and 4: Polygon) to create a quadrilateral which overlaps and contains the
same vertices of trapezoid ZOID.
7. Measure the area of the polygon using the
Area tool.
8. Find the lengths of b1, b2, and h so that we
can check the area given by your trapezoid area
formula in step five with the area value on the
screen. Use the Calculate tool to determine the
area using your formula in #5. Drag the vertices
around and observe if the formula continues to
match the area measurement. If the numbers do
not match, check your trapezoid formula with a
classmate.
9. Without making any additional measurements, drag point Z and explain how you know when figure
ZOID has been changed into a parallelogram. You may need to slide the entire figure to the right, or
make segment OI a little shorter so it fits on the screen.
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10. Once you have turned ZOID into a parallelogram, use the formula for the area of a trapezoid and
substitution to express the area of the parallelogram strictly in terms of b (only one variable is needed
because b1 = b2 now) , and h. Show how you derived the formula below.
AREA OF PARALLELOGRAM =
11. In Exploration 8-3, you estimated the area of
a large W on the coordinate plane. Now you can
find an exact value of the area using a process
called encasement.
This means a large rectangle will surround the W.
The inside of the rectangle will include a W, two
trapezoids, and five triangles. Since you have
been introduced to finding the area of a rectangle,
triangle, and trapezoid, you should be able to find
the area of the irregular polygon W.
Find the area of the W using encasement. The
origin of the hidden coordinate axes is placed on
the diagram. Show your work below.
EXPLORATION 8-6. PYTHAGOREAN THEOREM
Objective: Verify the Pythagorean Theorem.
1. CREATE A NEW DOCUMENT with the variable pythag.
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2. Construct a right triangle using perpendicular
lines. Create overlapping segments and hide the
lines.
The length of the smallest side will be called a;
The length of the middle-size side will be called
b;
and the length of the largest side will be called c.
3. Measure the lengths of a, b, and c.
4. Use the Calculate tool to find a2 + b2. You
will type in a^2+b^2 because the exponent is
represented by a caret, ^ . Also find c2.
5. Drag around the vertices of the right triangle and observe changes in your calculations.
6. State the formula for the Pythagorean Theorem below based on your conclusions from step 5.
7. A rational right triangle is a triangle where all the sides are rational numbers, and one of the angles is
a right angle. Find a rational right triangle such that the length of the hypotenuse is numerically equal
to the area of the triangle. Explain your analysis.
8. Find a rational right triangle such that the length of the hypotenuse is numerically equal to the area of
the triangle and the perimeter of the triangle is a prime integer. Explain your analysis.
EXPLORATION 8-7. CIRCUMFERENCE AND ARC LENGTH
Objective: Determine how the circumference and arc length relate to the radius of a circle.
1. CREATE A NEW DOCUMENT with the variable circumf.
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2. Create a large circle with a center near the
center of the screen. Measure the circumference
of the circle.
3. Use the Regular Polygon tool and create an
equilateral triangle, square, pentagon, and
hexagon on the circle with the circle's center.
Measure and label the perimeters of each of these
polygons and record their values, along with the
circle's circumference, using YOUR values in the
table below.
Regular Figure
Number of Sides
Perimeter/Circumference
Triangle
_____
__________
Square
_____
__________
Pentagon
Hexagon
_____
_____
__________
__________
Circle
???
__________
4. How many sides should an inscribed regular polygon have if its perimeter is the same as the circle's
circumference? Justify your answer below by explaining how the number of sides in an inscribed
regular polygon relates to the circumference of a circle.
5. Clear the screen. Create a new circle and radius in the middle of the screen. Measure the circle's
circumference and radius.
6. Use the Calculate tool and compute the circumference divided by two times the circle's radius (or
diameter) [I.e., circumference/(2*radius)]. Make sure that you include parantheses around the 2*r.
Modify the size of the circle and observe changes in your measurements and the status of the calculated
value. What value reoccurs in your calculation?
7. Use your observations in step six to write an equation below relating the circumference of a circle, c,
compared to its radius, r.
C=
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8. Create another radius of the circle and an arc
between the two points on the circle. Refer to
Exploration 6-1 to create an arc.
Measure the central angle of the sector which
contains the arc. Measure and label the length of
the arc using the Distance & Length tool (press
ENTER when the screen reads WHICH
OBJECT? and choose "THIS ARC").
9. Change the size of the circle until it has a
circumference very close to 10 cm.
10. Make the arc longer by moving the arc's endpoints further apart. Notice the change in the
measurement. Continue changing the arc until its length is equal to one-tenth the circumference of the
circle, about 1 cm. Record the measure of the central angle in the table below.
Change the arc length until the length of the arc is equal to one-quarter, one-half, and three-fourths of
the circle's circumference. Record the corresponding arc lengths and central angles in the table.
Fraction of Circle
1/10
1/4
1/2
3/4
Arc Length
__1 cm__
_________
_________
_________
Central Angle
_________
_________
_________
_________
Circumference
10 cm
10 cm
10 cm
10 cm
11. Use the table to help you find a pattern (formula) relating the arc length, central angle, and
circumference of the circle. Hint: Think of the portion of a circle the arc fills as a fraction of the full
circle. Reminder: A full circle measures 360˚ around.
12. Write a formula below for the arc length, L, in terms of the central angle, a, and the circumference
of the circle, C and any relevant numbers (like 360). Explain how you derived the formula.
L=
13. Check your formula using the numbers from a line in the table. Show below that your formula
works or does not work.
EXPLORATION 8-8. AREA OF A CIRCLE THROUGH DATA ANALYSIS
Objective: Collect data and determine the area of a circle in terms of the radius as a function.
1. CREATE A NEW DOCUMENT with the variable areacirc.
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2. Create a medium-sized circle near the center of the
screen. Measure and label the radius and area of the
circle as shown to the right.
3. Now you are ready to set the calculator up to store
data. Press F6 and choose “7: Collect Data” and
“2: Define Entry.” Move the cursor over the radius
measurement first until it reads THIS NUMBER, and
press ENTER . The calculator now recognizes that all
radius measurements will be stored in the first column
of the data table.
Move the cursor over the area measurement until it reads
THIS NUMBER and press ENTER . The calculator
now recognizes that all area measurements will be stored
in the second column of the data table.
Press ◊ and D to collect and store a data point.
Notice that the lower-left hand portion of the screen will
read DATA PLACED IN VARIABLE SYSDATA.
4. Access the stored data by pressing APPS and selecting 6:Data/Matrix Editor » 2:Open (shown
below to the left). In the variable option, use the arrow key to choose “sysdata” and press ENTER
twice (shown below to the right). You should now see the radius value stored in c1 and the area value
stored in c2. Note that the calculator automatically labels the columns for you.
Next, you will collect many data points by changing the radius of the circle and then storing each new
set of data in “sysdata” by pressing ◊ and D .
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5. You can view the transfer of information from
Geometry to the data editor by setting up a split
screen. Press
MODE and then F2 .
Use the arrows and ENTER key to set the Split
Screen to “LEFT-RIGHT,” with the Split 1 App
as Geometry and Split 2 App as Data/Matrix
Editor. Press ENTER and the Split Screen
should appear. You may need to press 2nd
and
APPS for both applications to appear on the screen. The 2nd and APPS sequence will also
toggle you back and forth between the geometry program and the data editor. You will need to stay on
the geometry side to continue collecting data.
6. On the geometry side, adjust the circle so the
radius and area measurements change. Press
◊ and D to collect and store another data
point. Notice that the data is immediately stored
in the “sysdata” editor on the right side of the
screen. Repeat this process with about 10 more
circles. Make sure you include very small and
very large circles in your data collection.
7. View the data on a graph by first setting up the
STAT PLOT menu in the Y= editor. Press
APPS and select 2:Y= Editor. Move the
cursor up to “Plot 1:,” and press ENTER . Use
the arrow key to set the x-value as data in column
c1 and the y-value as data in column c2 and press
ENTER twice. View the data on a graph by
pressing F2
and selecting 9: ZoomData.
8. Move back to the data table by pressing 2nd and
APPS . Use the F5 menu to find a best fitting
model. Choose a “calculation type,” set the x-value as
data in column c1 and the y-value as data in column c2,
and store the regression equation (RegEQ) to y1(x). Press
ENTER and the regression function should appear on
the screen.
9. Write an equation of the regression function below using three decimal place accuracy based on the
values of the regression equation.
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10. The leading coefficient in your equation is an approximation of a special number. Write an
equation relating the area of a circle (A) compared with its radius (r) using this special number (and not
the approximation).
11. Press Enter 2nd and Apps to view the graph of a model for the data. Explain in a sentence
below how the data relate to the curve.
EXPLORATION 8-8P. AREA OF TANGENT CIRCLES
Objective: Determine the constructions and solutions to problems related to the area of a circle.
1. CREATE A NEW DOCUMENT with the variable tancirc.
A circle is inscribed in a semicircle. That
means
the smaller circle passes only through the
semicircle's center and point on its
circumference.
2. Describe the steps to construct this diagram so that you can change its size, but still keep the smaller
circle inscribed in the semicircle.
3. What is the ratio of the area of the semicircle to the inscribed circle? Is this always true? After
computing the ratio on your screen, try to prove this conjecture by letting the radius of the inscribed
circle equal r. Show your work below.
4. CREATE A NEW DOCUMENT with the variable tancirc2.
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Two larger circles equal in radii pass through
each others centers. A smaller circle can be
created inside the overlapping region so that it
is tangent to the other circles. (Tangent means
that the circles touch each other, but do not
cross over each other, nor do they leave any
gaps.)
5. Describe the steps to construct this diagram so that you can change its size, but still keep the smaller
circle tangent to the larger circles.
6. What is the ratio of the larger circle to the smaller circle? Is this always true? After computing the
ratio on your screen, try to prove this conjecture by letting the radius of the smaller circle equal r. Show
your work below.
7. CREATE A NEW DOCUMENT with the variable tancirc3.
Along the diameter of a circle, you can
construct circles equal in radius length that are
tangent to each other and tangent to the circle.
That is, the circles touch each other, but do not
cross over each other, nor do they leave any
gaps.
8. Describe the steps to construct an even number of tangent circles so that you can change its size, but
still keep all of the circles tangent. How would your construction change if there were an odd number
of tangent circles?
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9. How does the combined area of all of the n smaller circles relate to the the area of the entire circle?
Give your answer in terms of n. Explain your reasoning.
10. After computing the ratio on your screen, try to prove this conjecture by letting the radius of the
smaller circles equal r. Show your work below.
EXPLORATION 9-5. FAMOUS PATHS IN GEOMETRY
Objective: Create three different geometric "designs" and relate them to materials studied in the
context of plane sections.
1. CREATE A NEW DOCUMENT with the variable path1
2. Draw a line segment AB. Draw another
segment with endpoint D on segment AB and
endpoint C above line segment AB.
3. Construct a perpendicular to segment AB
passing thru point D. Construct the perpendicular
bisector to segment CD. Mark the Intersection
Point, P, as shown to the left.
4. Drag point D back and forth along line
segment AB. What do you notice is the path of
point P? Take a guess if you're uncertain.
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5. A good way to check the validity of your guess
is to have the calculator Trace the path of point P
for you while point D moves back and forth.
Before we do this, though, let's hide the two lines
through P and segment CD to obtain a clearer
picture. Now press F7 and select 2: Trace On
/ Off and select point P. Finally, select animation
by pressing F7 and choosing 3: Animation.
Next, click on the hand tool, and drag D toward
B. You will see a spring form. (The longer the
spring when you release the hand tool, the faster point D
will be animated along the segment.) Hit escape to
stop. What kind of curve (or portion of curve)
seems to have been generated?
6. CREATE A NEW DOCUMENT with the variable path2.
7. Draw a circle with center C. Place a point B
inside the circle. Place point A on the circle (use
Point on Object).
p. 83
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8. Construct segment AB. Construct the perpendicular bisector of segment AB. Draw a line
through points A and C. Name the intersection
point of line AC and the perpendicular bisector of
AB point P. See the diagram below.
9. Drag point A along the circle. What do you
think will be the path of point P? Guess if you're
not sure.
10. To test your guess, we will have the calculator Trace the path of point P for you while point
A moves along the circle. Before we do this let's
hide some non-essential clutter as we did earlier.
Hide the two lines and the segment so your screen
matches the picture to the right.
Select point P to be traced. Animate point A.
Recall that you do not hit enter to select point A - you use the hand tool and drag it to one side
(you'll see the spring). Hit escape to stop.
What kind of curve seems to have been
generated?
11. Choose Undo. Your curve should be gone. For the third and final path, simply drag point B to a
point outside the circle.
12. When you move point A along the circle,
what do you think seems to be the path of point
P?
As before, guess if you're not sure.
13. Well, let's get a better look. Trace point P
and animate A once again. What kind of curve
seems to have been generated?
EXPLORATION 10-6. CREATING A TOOLBAR TO REFERENCE FORMULAS
Objective: Write a program that create a toolbar to reference lateral area, surface area, and volume
formulas.
1. Enter a New Program
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Press APPS
and choose 7:Program Editor » 3:New. Save the file in the main directory folder your
name as the variable, and press ENTER
twice as shown below.
2. You are now in the programming entry screen
when your screen matches the diagram to the
right (replace your name with george).
The program that you are about to create will set
a custom title bar where you can access geometry
formulas.
3. Press F2 Control and choose 7:Custom...EndCustm. The program contains a block of
statements between "Custom" and "EndCustm." Each of those statements is either a "Title" (of a pulldown menu) or "Item" (entry in a pull-down menu) statement.
4. Create each Title by pressing F3 I/O and
choose 1:Dialog » 7:Title (or type the word
Title), and then type "Prism". Press ENTER to
move to the next line.
Repeat this procedure for Cylinder, Pyramid,
Cone, and Sphere, as shown to the right.
5. Check the status of your program by pressing ◊ Q
to return to the home screen. Type the name
of the program with parenthesis, yourname(), press ENTER
shown below to the left) and press 2nd 3
until the screen reads "Done," (as
to activate the custom toolbar (as shown below to the
right). Notice how the toolbar changes at the top of the screen. Press 2nd 3
original title bar.
to return to the
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6. The next step will include the items in the pull down menu. Press APPS and choose 7:Program
Editor »1:Current to return to your program. Move the cursor to the right of "Prism" and press
ENTER to create a new line as shown below to the left.
7. Create a listing by pressing
F3 I/O
and choose 1:Dialog » 8:Item (or type the word Item).
Type "LA = ph." Press ENTER to advance to the next line and make the lists SA = LA + 2B and
V=Bh, as shown above to the right.
8. Go back to the main screen, re-enter the
program name, and look at the custom menu.
Press F1 to view the prism formulas you have
just entered.
9. Your goal is to finish the toolbar until there are
formulas under each menu. Your formulas
should be as specific as possible. For example,
the volume of a cylinder should not be V=Bh, but
instead V=πr^2h, because you can substitute πr^2
for B, the area of the base.
Go back to the program and start with the next Item under the Cylinder Title. When you have
completed the program, try it out. There should be a total of 14 formulas.
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EXPLORATION 11-6. DISTANCE FORMULA
Objective: Verify and prove the distance formula.
1. CREATE A NEW DOCUMENT with the
variable distform.
2. Use the Format tool to turn on the coordinate
axes to a rectangular grid.
3. Plot a point in quadrant II and another point in
quadrant IV. Measure and label the distance
between the points.
4. Measure the coordinates of the two points
Press F6 and choose 5:Equation &
Coordinates, move the pencil on top of one of
the points until the screen reads
COORDINATES OF THIS POINT, press
ENTER and repeat the process with the second
point.
5. Use the Calculate tool to verify the measured
distance using the distance formula:
d = (x2-x1)2 + (y2-y1)2
The calculation strip should compare with the
diagram to the right. The locations of the values
of a, b, c, and d on the screen, respectively
correspond with x2, x1, y2, and y1 in the formula.
Press ESC to return to the pointer. Drag the
points around the screen. If the distance
measurement continues to equal the calculated
value, then you have verified the distance
formula!
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6. Connect the two points with a segment. Use
horizontal and vertical segments to draw a picture
of a right triangle shown to the right. Find the
coordinates at the right angle.
Do not move any of the vertices of the triangle
because then it will no longer be a right triangle.
7. Use the coordinates to calculate the lengths of the legs of the right triangle WITHOUT measuring
Distance & Length. You may use the calculator for addition and subtraction purposes only. Show
your work below.
8. Use these calculations and the hypotenuse to verify the Pythagorean Theorem. Show your work
below.
9. Suppose the triangle had coordinates (x2,y1),
(x1,y1), and (x2,y2), as shown to the right.
Explain below how the Pythagorean Theorem can
be used to prove the distance formula:
d=
(x2-x1)2 + (y2-y1)2
EXPLORATION 11-7. EQUATIONS OF CIRCLES
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Objective: Determine the relationship between the equation of a circle to its center and radius.
1. CREATE A NEW DOCUMENT with the variable eqcirc.
2. Use the Format tool to turn the coordinate
axes to rectangular and the grid to on. Change
the axes unit from 0.5 to 2. Recall calc. must say
THIS UNIT and then drag it closer to the origin.
3. Create a circle with center at the origin and a
radius of two as shown to the right. Make sure
the radius point lands on the grid.
3. Find the equation of the circle
Press F6 and choose 5:Equation & Coordinates, move the pencil on top of the circle until the
screen reads EQUATION OF THIS CIRCLE, and press ENTER .
*The equation should read x2 + y2 = 22, which is really x2 + y2 = 4.
4. Drag the radius of the circle until it equals 0, 4, and 6. Record the equations in the table below.
Center
(0,0)
(0,0)
(0,0)
Radius Equation on Calculator
Simplified Form
0
__________________
__________________
2
x2 + y2 = 22
x2 + y2 = 4
4
__________________
__________________
(0,0)
6
__________________
__________________
5. Use the information in the table to describe below how the radius affects the equation of a circle
centered at the origin.
6. Delete the circle. Now you will determine how
the location of the center affects the equation of a
circle.
Create a new circle with radius equal to 2 and
center at coordinates (4,2), as shown to the right.
Make sure the radius point lands on the grid.
7. Find the equation of the circle. You may need
to drag the measurement back on the screen. The
equation should read (x-4)2 + (y-2)2 = 22, which
is really (x-4)2 + (y-2)2 = 4.
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8. Drag the center of the circle until it has coordinates (-4,4), (-6,-2), and (10,-4). Record the equations
in the table below.
Center
(4,2)
(-4,4)
(-6,-2) 2
Radius Equation on Calculator
Simplified Form
2
2
2
2
(x-4) + (y-2) = 2
(x-4)2 + (y-2)2 = 4
2
__________________
__________________
__________________
__________________
9. Use the information in the table to describe below how the center affects the equation of a circle
centered at the origin.
10. State the standard equation of a circle with center (h, k) and radius r. Your equation should not
include numbers and should include the variables x, y, h, k, and r.
11. Demonstrate algebraically that your answer in #10 is indeed correct. Hint: Use the distance formula and
the fact that the radius is the distance from the center to a generic point on the circle (x,y).
12. Use the standard equation of a circle to complete the missing information in the table below. Use
the calculator to check your answers. You will need to change the axes unit to check some of your
answers.
Center
A. (8,4)
B. (-2,0)
Radius Simplified Form
6
____________________
_____
(x+2)2 + y2 = 16
C. _____
D. _____
10
_____
(x-10)2 + (y+5)2 = 100
x2 + (y-50)2 = 2500
EXPLORATION 11-8. MEANS AND MIDPOINTS
Objective: Determine the relationship between the equation of a circle to its center and radius.
1. CREATE A NEW DOCUMENT with the variable mnmid.
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2. Create ∆CUB. Let H be the midpoint of CU
and I be the midpoint of CB. Create segment HI
as shown to the right. HI, the segment connecting
the midpoint of two sides of a triangle, will be
referred as the midline to the base UB.
3. Measure the slopes of midline HI and base UB.
To measure the slope of UB you will need to create an
overlapping segment. The calc. cannot determine slopes of
sides of polygons. It can only find slopes for segments.
4. Drag around the vertices and observe the changes in measurement. Use a complete sentence to state
below the relationship between a midline and its corresponding base without using the word slope.
5. Measure the lengths of midline HI and base UB. Drag around the vertices and observe the changes
in measurement. Use a complete sentence to state below the relationship between the length of a
midline and its corresponding base.
6. Clear the screen. Use the Format tool to turn
on the coordinate axes to a rectangular grid.
7. Create segment CT with endpoints at (3, 1) and
(1,0). Create the midpoint M of segment CT.
Measure, label, and organize the coordinates of
the points, as shown to the right.
8. Drag around point T around the screen and
observe the relationship between the midpoint M
and the endpoints of segment CM.
9. Suppose the coordinates of C was (x1,y1), and
T was (x2,y2). What would be the coordinates of
the midpoint M? Express your answer in terms
of x1, x2, y1, and y2. State your answer and
explain your reasoning below.
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10. Use your formula to complete the missing information in the table below. Use the calculator to
check your answers.
endpoint C
A. (1,1)
B. (-1.5,0.5)
C. ________
D. (-3,-1)
midpoint M
________
________
(1.75,-1)
(-4, -1.5)
endpoint T
(3,0)
(1.5,-1.5)
(1.5, 0)
________
11. Clear the screen. Construct a triangle and
its medians as shown to the right. The
intersection of the medians of a triangle is
called the centroid. How do the coordinates of
the centroid relate to the coordinates of the
vertices of the triangle? Write an equation to
describe this relationship. Explain how you
determined this result..
EXPLORATION 11-8P. MIDPOINTS AND AREAS
Objective: Determine the relationship between the area of a triangle and rectangle related to
midpoints.
1. CREATE A NEW DOCUMENT with the variable midarea.
2. Construct a triangle connecting the three
midpoints of the sides, as shown to the right.
Triangle DEG is called the medial triangle.
Use the polygon tool to create a triangle that
overlaps triangle DEF.
3. What is the ratio of the areas of the larger triangle ABC compared to the medial triangle DEF? Is this
property always true? Explain why or why not. Prove your case.
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4. What is the ratio of the perimeters of the larger triangle ABC compared to the medial triangle DEF?
Is this property always true? Explain why or why not. Prove your case.
5. How will you results to steps three and four change if you examined the medial polygons inside
different polygons with n sides? Explain your analysis.
6. CREATE A NEW DOCUMENT with the variable midarea2.
7. Construct triangle GOR. Construct
midpoints D and E of segments OR and GO,
respectively. Construct rectangle MATH (make
sure that you hide any lines used in the
construction). Use the polygon tool to create a
polygon on top of GOR and a polygon on top
of MATH.
8. Measure the area of triangle GOR and the
area of rectangle MATH. How do they relate?
Why is this true? (Hint: think of the midline
ED). Explain your reasoning.
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EXPLORATION 12-1. THE TRANSFORMATION SK
Objective: Determine the effects of a segment under a transformation Sk.
1. CREATE A NEW DOCUMENT with the variable skthm.
2. Clear the screen. Use the Format tool to turn
on the coordinate axes with rectangular grid.
3. Create segment MN with M near (1.00,0.25)
and N at (-0.50,0.50). Measure, label, and move
the coordinates until they correspond with the
diagram to the right.
4. The next step will be to create a transformation
of segment MN under Sk, where k is the
magnitude of the transformation.
Create a horizontal segment with length 2 cm in the upper left hand portion of the screen, beginning at
a grid point, as shown below in the diagram to the right. Measure the segment and label it with k=.
This segment and measurement will represent the magnitude of transformation, k.
5. Create a dilation of segment MN
Press F5 and choose 3:Dilation, move the
pencil on top of segment MN until the screen
reads DILATE THIS SEGMENT, press
ENTER , move the pencil to the origin until the
screen reads WITH RESPECT TO THIS
POINT, press ENTER , move the pencil to the
measurement 2.00 cm (point at the #, not the segment)
until the screen reads USING THIS FACTOR,
and press ENTER .
6. Press ESC to return to the pointer, and don't bother labeling the endpoints of the image M' and N'
as shown above to the right. That's just done for your reference.
7. Measure, label, and organize the coordinates of M' and N'.
8. Adjust the value of k to equal 1, 2, and 3, by changing the size of the segment in the upper left hand
portion of the screen. Record the coordinates of the image in the table below.
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Magnitude (k)
1
M
(1, 0.25)
M'
N
N'
_________
(-0.50,0.50)
_________
2
3
(1, 0.25)
(1, 0.25)
_________
_________
_________
_________
(-0.50,0.50)
(-0.50,0.50)
9. Use the table to explain below how the coordinates of a point change under a transformation Sk.
Use k in your answer. Be precise.
10. Measure the length of segment MN and M'N'. Adjust the value of k by changing the size of the
segment in the upper left hand portion of the screen, and compare the length of MN to its image M'N'.
Explain below the relationship between the length of a segment and its image under a transformation
Sk.
11. Measure the slope of segment MN and M'N'. Adjust the value of k and compare the slope of MN
to its image M'N'. Explain below the relationship between the slope of a segment and its image under
a transformation Sk.
10. Use your conclusions in this exploration to complete the missing information in the table below.
Use the calculator to check your answers.
Magnit.
k
Coord.
M
1/2
(1.00,.25)
(1.00,.25)
4
Coord.
M'
Coord.
N
Coord.
N'
(-.5,.5)
(2.00,.50)
(-.5,.5)
(2.25,3.0)
(-.5,-.5)
Length
of M'N'
1.52
(0,2)
(4,0)
Length
of MN
Slope
of M'N'
-0.17
1.26
-0.75
6.32
(-1.5,-1.5)
Slope
of MN
5.86
-0.33
1.19
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EXPLORATION 12-2. SIZE CHANGES
Objective: Explore the transformation of a figure under a size change with varying magnitude.
1. CREATE A NEW DOCUMENT with the variable size.
2. Create a small horizontal segment in the upper
left hand corner of the screen to represent the
adjustable magnitude k. Measure the length of
this segment and label the measurement with k=.
Create point B in the lower left corner of the
screen to represent the center of dilation (or size
change). Create a small quadrilateral QUAD near
the left-middle of the screen with the Polygon
tool, as shown to the right.
3. Use the Dilation tool to perform a size change
of magnitude k on the quadrilateral about point B.
You do not need to label the image Q'U'A'D' as
shown to the right, but you must recognize which
is the image of which.
4. Modify the value of k until the image Q'U'A'D'
becomes smaller, the same size, and larger, than
the preimage QUAD. Complete the blanks
below.
The image contracts when ____ < k < ____
The image remains the same size when k = ____
The image expands when k > ____
5. Measure and label the distance from B to D, and B to D'. Modify the value of k and observe the
changes in the measurements.
6. Predict and write an equation below using multiplication relating BD, B'D', and k, without using any
numbers.
7. Use the Calculate tool and input the side of the formula with the k and BD. Modify the value of k
and compare the formula's output with the measurement of B'D'. If the numbers are the same, then
your formula is correct! If the numbers are not the same, check your formula with a classmate.
8. Measure and label the length of segments AD and A'D'. Modify the value of k and observe the
changes in the measurements.
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9. Predict and write an equation below using multiplication relating AD, A'D', and k, without using any
numbers.
10. Use the Calculate tool and input the side of the formula with the k and AD. Modify the value of k
and compare the formula's output with the measurement of A'D'. If the numbers are the same, then
your formula is correct! If the numbers are not the same, check your formula with a classmate.
EXPLORATION 12-3. PROPERTIES OF SIZE CHANGES
Objective: Examine properties related to collinearity and angle measurement under a size change with
varying magnitude.
1. OPEN the document with the variable size
contained in your folder.
2. Delete all of the measurements and
calculations, except for the value of k, as shown
to the right.
3. Check the collinearity of the dilation point, a
vertex, and its image. Press F6 and choose
8:Check Property » 1:Collinear, move the
pencil on top of point B until the screen reads
THIS POINT, press ENTER , move the pencil
on top of point D until the screen reads THIS
POINT, press ENTER , move the pencil on top
of point D' until the screen reads THIS POINT,
and press ENTER .
A dotted box will appear on the screen, as shown
to the right. Drag the box to the right side of the
screen and press ENTER . The screen will tell
you whether the points are collinear or not
collinear.
4. Check the collinearity of the other three sets of points (BQQ', BAA', BUU'). Drag around the
vertices of QUAD to see if the collinearity changes. Use your observations to describe a property of a
size change below without using any form of the word collinear (or explain what collinearity means).
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5. Delete the collinearity outputs on the screen. Measure and label all of the angles in QUAD and
Q'U'A'D'. Drag around the vertices of QUAD and compare the corresponding angle measurements.
Use your observations to describe another property of a size change below.
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EXPLORATION 12-4. PROPORTIONS
Objective: Use size changes to learn relationships about proportions.
1. OPEN the document with the variable size
contained in your folder.
2. Delete all of the measurements and
calculations, except for the value of k, as shown
to the right.
3. Measure and label the lengths of the sides of
both polygons.
4. Use the Calculate tool to compute the ratios
Q'U' A'U' D'A'
D'Q'
QU , AU , DA , and DQ
5. Modify the value of k and explain below how the ratios compare with each other and the magnitude
value, k.
DQ
D'Q'
6. Proportions represent two equal ratios. For example, you can show that DA = D'A' . Use the
Calculate tool to find at least two more proportions. Record those proportions below.
7. Use the Calculate tool to compute DQ*D'A' and DA*D'Q'. Modify the value of k and observe the
changes in the calculations. Use the proportion in step 6 and explain below how these computations
verify the Means-Extremes Property.
EXPLORATION 12-5. SIMILARITY
Objective: Verify properties of similarity through transformations and measurement.
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1. OPEN the document with the variable size
contained in your folder.
2. Delete all of the measurements and
calculations, except for the lengths of the sides in
QUAD, and the value of k, as shown to the right.
3. Adjust the value of k so that the image
Q'U'A'D' is to the left of the center of the screen.
4. Create a vertical line to the right of Q'U'A'D'
and Reflect Q'U'A'D' over the line.
Label the reflected polygon Q"U"A"D", and Hide
Q'U'A'D', as shown the right.
5. Measure the corresponding angles, and
corresponding proportions
Q"U" A"U" D"A"
D"Q"
,
,
,
and
QU
AU
DA
DQ . Modify the
value of k and observe the changes in
measurement and calculation.
6. Explain below why QUAD and Q"U"A"D" are similar figures under a composite of a size change
and reflection.
7. CREATE A NEW DOCUMENT with the
variable similar.
8. Use perpendicular lines to construct right
triangle ABC, with right angle B. Construct an
altitude from B to hypotenuse AC, as shown to
the right. Let the altitude intersect the hypotenuse
at point D. Drag around the vertices of the
triangle to ensure the right angle and altitude keep
their form.
9. Use the Measure tool to find similar triangles in the diagram (some are overlapping). State and
draw below at least two pairs of similar triangles from the diagram. Justify why the triangles are
similar.
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EXPLORATION 12-6. THE FUNDAMENTAL THEOREM OF SIMILARITY
Objective: Show the relationship between the magnitude of a size change (k) compared to the
perimeter and area of two similar figures.
1. OPEN the document with the variable size
contained in your folder.
2. Delete all of the measurements and
calculations, except for the value of k, as shown
to the right.
3. Measure the perimeters of the two
quadrilaterals.
4. Predict and write an equation below using
multiplication relating the perimeter of QUAD,
the perimeter of Q"U"A"D", and k, without using
any numbers.
5. Use the Calculate tool and input the side of the formula with the k and the perimeter of QUAD from
your answer in #4. Modify the value of k and compare the formula's output with the measurement of
the perimeter of Q"U"A"D". If the numbers are the same, then your formula is correct! If the numbers
are not the same, check your formula with a classmate.
6. Measure the areas of the two quadrilaterals.
7. Predict and write an equation below using multiplication relating the area of QUAD, the area of
Q"U"A"D", and k, without using any numbers. Hint: Area uses square units.
8. Use the Calculate tool and input the side of the formula with the k and the area of QUAD. Modify
the value of k and compare the formula's output with the measurement of the area of Q"U"A"D". If the
numbers are the same, then your formula is correct! If the numbers are not the same, check your
formula with a classmate.
EXPLORATION 13-1. THE SSS SIMILARITY THEOREM
Objective: Show that three sets of equal ratios will produce similar triangles
1. CREATE A NEW DOCUMENT with the variable simsss.
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2. Create overlapping triangles CAT and FAR
with common vertex A and segment FR parallel
to segment CT, as shown to the right. You will
have to construct a parallel line FR, then make an
overlapping segment FR. Hide the parallel line
(but
not the segment) that is necessary in the
construction.
Drag around the vertices of ∆CAT to ensure that
FR remains parallel to CT.
3. Measure and label the lengths of all of the
sides of both triangles.
CA CT
TA
4. Use the Calculate tool to compute the following ratios: FA , FR , and RA . Drag around point F
or point R and observe the changes in your calculations. What can you conclude about these ratios?
5. Without measuring, explain below why the corresponding angles in ∆CAT are equal to those in
∆FAR.
6. Given that three corresponding ratios are equal, is this enough to verify that ∆FAR~∆CAT? Explain
why or why not below.
EXPLORATION 13-2. THE AA AND SAS SIMILARITY THEOREMS
Objective: Show that the AA and SAS conditions produce similar triangles
1. CREATE A NEW DOCUMENT with the variable simsas.
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2. Construct triangles ABC and EBD by creating
two parallel lines and two intersecting segments
between them, as shown to the right.
3. Create overlapping segments AC and DE. Hide
the two lines.
4. Without measuring, state below three pairs of
congruent corresponding angles. Explain why the
pairs of angles are congruent.
5. Measure and label the lengths of the sides in both triangles. Use the Calculation tool to compute
AB AC
CD
BE , DE , and BD .
6. Drag the vertices of the triangles around and observe the changes in ratios. Given that three
corresponding angles are congruent (AAA), is this enough to verify that ∆ABC~∆EBD? Explain why
or why not below.
7. Is it possible to show ∆ABC~∆EBD with only two pairs of congruent corresponding angles (AA)?
Explain why or why not below.
8. Clear the screen. Construct two dotted
concentric circles (same center) with center A.
Create two non-overlapping isosceles triangles
ABC and ADE, whose legs are equal to the radii
of their respective circles, as shown to the right.
9. Measure and label < DAE and < CAB. Drag
point E around the circle until both angles have
the same measurement.
10. Without measuring, explain below why
AD
AE
AC = AB .
p. 103
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reproduced without permission (phone: 847-486-4683 or e-mail: [email protected]).
11. Measure and label DE, CB, EDA, DEA, ACE, and ABC.
DE
Use the Calculate tool to compute CB .
12. Modify the larger circle's size and observe the changes in corresponding ratios and angle
measurements. Given that two corresponding ratios are equal and an included pair of angle is
congruent (SAS), is this enough to verify that ∆ABC~∆ADE? Explain why or why not below.
EXPLORATION 13-2P. TRIANGLES IN A TRAPEZOID
Objective: Find the relationship between the area of triangles in a trapezoid.
1. CREATE A NEW DOCUMENT with the variable tritrap.
2. Construct a trapezoid, diagonals, and
intersection of diagonals as shown to the right.
3. Which triangles in the diagram are similar?
Explain how you know.
4. Find an equation relating the ratio of the bases of the trapezoid, the area of triangle CED and the area
of triangle AEB. Explain how you determined this result.
5. Which triangles in the diagram have the same area? Explain why this is true.
EXPLORATION 13-3. THE SIDE SPLITTING THEOREM
Objective: Verify the side-splitting theorem and its converse
p. 104
The following materials were produced by the Glenbrook South High School mathematics department and may not be
reproduced without permission (phone: 847-486-4683 or e-mail: [email protected]).
1. OPEN the file simsss in your folder.
2. Delete all of the measurements and
calculations, except for the lengths of FA and
RA, as shown to the right.
3. Measure the lengths of segments FC and RT.
Use the Calculate tool to compute the ratios
AF
AR
FC and RT .
4. Drag around point F or point R and observe changes in the calculated ratios.
5. If a parallel segment (AG) splits two sides of a triangle, what can you conclude about the ratios of
the split sides? Explain below.
6. CREATE A NEW DOCUMENT with the variable sidespl.
7. Create overlapping triangles LUH and AUG,
as shown to the right. Segment AG should not be
drawn parallel to segment LH. Create an
overlapping segment LH.
8. Measure and label the slopes of segments AG
and LH, and the lengths of segments UA, AL,
UG, and GH.
9. Use the Calculate tool and compute the ratios
AL
GH
UA and UG .
AL
GH
10. Drag point A or point G until UA = UG . Observe the changes in the slope of AG and LH. If a
segment (AG) splits two sides of a triangle into equal proportions, what can you conclude about that
segment and the base of the triangle (LH)? Explain why below.
INDEX
p. 105
The following materials were produced by the Glenbrook South High School mathematics department and may not be
reproduced without permission (phone: 847-486-4683 or e-mail: [email protected]).
Operation
AA similarity
AAS postulate
acute triangle
alternate interior angles
altitude
trapezoid
triangle
angle(s)
alternate interior
bisector
central
corresponding
create
exterior
interior
negative
size change
prop
straight
sum in polygon
unequal
vertical
arc
arc length
area
circle (formula)
circle (measure)
irregular poly
lateral
parallelogram
rectangle
surface
trapezoid
triangle (form)
triangle (meas)
ASA postulate
axes
coordinate
unit
First seen in
Exploration
13-2
7-2
8-4
5-4
8-5
8-4
5-4
3-1
8-7
3-6
0-9
7-9
7-9
4-5
12-3
7-9
5-7
7-9
3-3
6-1
8-7
8-8
0-8
8-3, 8-5
10-6
8-5
8-2
10-6
8-5
8-4
0-10
7-2
1-3
1-3
Operation
base of triangle
bisector
angle
perpendicular
calculate
central angle
change angle
check property
circle
area (formula)
area (measure)
circumference
concentric
diameter
equation
radius
circumference
formula
measure
clear screen
collinearity
comment
condition, SSA
congruent triangles
contrast
collecting data
comment
concentric circles
congruent triangles
coordinate
measure
plane
corresponding angles
cross
darken the screen
data collection
delete
diagonal
kite
parallelogram
diameter
dilation
distance
formula
measure
minimum
p. 106
First seen in
Exploration
8-4
3-1
3-8
0-10
8-7
0-9
2-8
8-8
0-8
0-8
13-2
8-7
11-7
0-8
8-7
0-8
0-6
12-3
0-8
7-5
7-2
0-1
8-8
0-8
13-2
7-2
11-6
4-2
3-6
0-4
0-1
8-8
0-6, 0-8
6-4
7-7
8-7
12-1
11-6
0-8
4-3
The following materials were produced by the Glenbrook South High School mathematics department and may not be
reproduced without permission (phone: 847-486-4683 or e-mail: [email protected]).
INDEX (page 2)
p. 107
The following materials were produced by the Glenbrook South High School mathematics department and may not be
reproduced without permission (phone: 847-486-4683 or e-mail: [email protected]).
Operation
document, new
dotted
drawing, perspective
edit
label
numerical
encasement
equation
circle
line
regression
equilateral triangle
error
delete
wrong key
explicit formula
exterior angles
folder, new
format
formula
arc length
area, similarity
area of circle
area of rectangle
area of triangle
circumference
distance
equat of circle
equation of line
explicit
lateral area
line, equation
midpoint
perim, similarity
Pythag Thm
recursive
slope
sum of poly <s
reg poly rotation
reg poly perim
surface area
volume
fund theorem similarity
First seen in
Exploration
0-4
4-5
1-5
0-5
7-6
8-5
11-7
1-3
8-8
6-2
0-6
0-2
5-7
7-9
0-3
1-3
8-7
12-6
8-8
8-2
8-4
8-7
11-6
11-7
1-3
5-7
10-6
1-3
11-8
12-6
8-6
5-7
3-6
5-7
6-6
8-1
10-6
10-6
12-6
Operation
glide reflection
grid
height
trapezoid
triangle
hide
HL postulate
interior angle
intersecting lines
intersection point
irregular polygon, area
isosceles trapezoid
isosceles triangle
k, magnitude
kite
label
line
measurement
point
lateral area formulas
legs, isos. trapezoid
lighten the screen
line(s)
create
equation
horizontal
label
oblique
parallel
perpendicular
vertical
macro
magnitude
means-extremes
measure
angle
area
circumference
coordinates
distance
length
perimeter
radius
segment length
slope
median
p. 108
First seen in
Exploration
4-7
1-3
8-5
8-4
1-5
7-5
7-9
0-7
0-7
8-3, 8-5
6-5
6-2
12-1
6-4
3-6
0-8
0-5
10-6
6-5
0-1
0-7
1-3
1-3
3-6
0-7
1-5, 3-6
3-7
1-3
4-7
12-1
12-4
0-9
0-8
0-8
11-6
0-8
0-8
0-10
0-8
0-8
3-6
6-2
The following materials were produced by the Glenbrook South High School mathematics department and may not be
reproduced without permission (phone: 847-486-4683 or e-mail: [email protected]).
INDEX (page 3)
Operation
midline
midpoint
connection
create
formula
minimum distance
move
angle
label
measurement
point
mystery quadrilaterals
negative angle
new
document/sketc
h
folder
program
numerical edit
oblique lines
obtuse triangle
orthocenter
parallel lines
parallelogram
area
construct
diagonals
properties
perimeter
regular polygon
triangle
perpendicular
bisector
bisector theorem
lines
perspective drawing
plane, coordinate
First seen in
Exploration
11-8
Operation
point
creating
intersection
label
midpoint
move
on object
on perp. bisector
vanishing
11-8
2-4
11-8
4-3
0-9
0-5
0-9
0-5
6-M
4-5
0-4
0-3
10-6
3-2, 6-3
8-4
8-4
1-5, 3-6
8-5
6-3
7-7
7-7
8-1
0-10
3-8
5-5
3-7
1-5
1-3
First seen in
Exploration
pointer
polygon
create
special quad
postulate
AAS
ASA
HL
SAS
SsA
SSS
press the wrong key
program, new
properties, size change
proportions
Pythagorean Theorem
quadrilateral
create
mystery
radius
ratio
ray
rectangle
area
construct
polygon tool
recursive formula
0-5
0-7
0-5
2-4
0-5
0-9
5-5
1-5
0-4
4-7
8-2
7-2
7-2
7-5
7-2
7-5
7-2
0-2
10-6
12-3
12-4
8-6
6-3
6-M
0-8
12-4
0-9
8-2
6-3
8-2
5-7
p. 109
The following materials were produced by the Glenbrook South High School mathematics department and may not be
reproduced without permission (phone: 847-486-4683 or e-mail: [email protected]).
p. 110
The following materials were produced by the Glenbrook South High School mathematics department and may not be
reproduced without permission (phone: 847-486-4683 or e-mail: [email protected]).
INDEX (page 4)
p. 111
The following materials were produced by the Glenbrook South High School mathematics department and may not be
reproduced without permission (phone: 847-486-4683 or e-mail: [email protected]).
Operation
First seen in
Exploration
reflect
application
figure
glide
point
symmetric
regression equation
regular polygon
create
perimeter
remove object(s)
revise a label
rhombus
right triangle
rotations
of a figure
symmetry
SAS postulate
SAS similarity
scalene triangle
segment
select object
show
side splitting theorem
sides, unequal
similarity
AA
fund theorem
introduction
SAS
SSS
size change
create
properties
Sk, transformation
slides
slope
formula
identify
measure
split screen
square
SSA condition
SsA postulate
SSS postulate
SSS similarity
4-3
4-4
4-7
4-1
6-1
8-8
1-5, 6-7
8-1
0-6
0-5
6-3
8-4
3-2, 4-5
6-6
7-2
13-2
6-2
0-8
0-6, 4-7
8-4
13-3
7-9
13-2
12-6
12-5
13-2
13-1
12-2
12-3
12-1
4-6
3-6
1-3
3-6
8-8
1-5, 6-3
7-5
7-5
7-2
13-1
Operation
sum of angles
formula
triangle
polygon
surface area formulas
symmetry
reflection
rotation
straight angle
swimmer
sysdata
tessellations
transformation Sk
translations
transversal
trapezoid
area
construct
triangle(s)
acute
altitude
area (formula)
area (measure)
base
congruent
creating
equilateral
height
inequality
isosceles
obtuse
orthocenter
right
scalene
tessellations
translation
transversal
unequal angles/sides
unit, axes
vanishing point
vector
vertical angles
volume formulas
wrong key error
y-intercept
First seen in
Exploration
5-7
6-1
5-7
10-6
6-1
6-6
7-9
4-3
8-8
7-6
12-1
4-6
3-6
8-5
6-5
8-4
8-4
8-4
0-10
8-4
7-2
0-10
6-2
8-4
2-7
6-2
8-4
8-4
8-4
6-2
7-6
4-6
3-6
7-9
1-3
1-5
4-6
3-3
10-6
0-2
1-3
p. 112
The following materials were produced by the Glenbrook South High School mathematics department and may not be
reproduced without permission (phone: 847-486-4683 or e-mail: [email protected]).
VARIABLE INDEX
Variable Name
Exploration
Topic
altint
angbis
angles
areacirc
bounce
capture
circ
circumf
compref
compref2
congru
congtri
conject
coordin
distform
eqcirc
eqline
extang
glideref
inequal
irrarea
isostri
kite
lines
midarea
midarea2
midpoint
mnmid
mystery (folder)
optimal
optimal2
5-4
3-1
0-9
8-8
7-7P
5-5P
0-8
8-7
4-4
4-5
5-2
7-2
2-8
4-2
11-6
11-7
1-3
7-9
4-7
2-7
8-3
6-2
6-4
0-7
11-8P
11-8P
2-4
11-8
6-M
8-2P
8-2P
Alternate interior angles
Angle bisector
Create, measure, and change an angle
Area of a circle through data analysis
Bouncing off the walls
Capture the flag
Create, measure, and change a circle
Circumference and arc length
Translations: Composing reflections over parallel lines
Rotations: Composing reflections over intersecting lines
Congruence and equality
Congruent triangles
Testing conjectures
Reflecting figures in a coordinate plane
Distance formula
Equations of circles
Equations of lines
Exterior angles
Glide reflections
Triangle inequality
Areas of irregular polygons
Isosceles and equilateral triangles
Constructing a kite
Create two intersecting lines
Midpoints and areas
Midpoints and areas
Midpoints
Means and midpoints
Mystery quadrilaterals
Optimal quadrilaterals
Optimal quadrilaterals
Continued on next page
p. 113
The following materials were produced by the Glenbrook South High School mathematics department and may not be
reproduced without permission (phone: 847-486-4683 or e-mail: [email protected]).
VARIABLE INDEX (page 2)
p. 114
The following materials were produced by the Glenbrook South High School mathematics department and may not be
reproduced without permission (phone: 847-486-4683 or e-mail: [email protected]).
Variable Name
parallel
path1
path2
pararect
perpbis
perpline
perspect
pgram
point
polysum
ptperp
pythag
rectarea
reflect
refsym
regperim
regpoly
rotate
rotsym
shark
sidespl
similar
simsas
simsss
Exploration
3-6
9-5
9-5
6-3
3-8
3-7
1-5
7-7
0-4, 0-5
5-7
5-5
8-6
8-2
4-1
6-1
8-1
6-7
3-2
6-6
6-2P
13-3
12-5
13-2
13-1
13-3
size
12-2
12-3
12-4
12-5
12-6
skthm
12-1
ssahl
7-5
swimmer
4-3
tancirc
8-8P
tancirc2
8-8P
tancirc3
8-8P
tessell
7-6
traparea
8-5
trapez
6-5
treasure
3-8P
triangle
0-10
triarea
8-4
trirec
8-4P
tritrap
13-2P
vector
4-6
vertang
3-3
your name (pgm) 10-6
Topic
Slope and parallel lines
Famous curves
Famous curves
Constructing parallelograms
Perpendicular bisector
Perpendicular lines
Drawing in perspective
Properties of parallelograms
Create, label, and move points
Sum of angles in a polygon
Perpendicular bisector theorem
Pythagorean Theorem
Area of a rectangle
Reflecting points
Reflection-symmetric figures
Perimeter of a regular polygon
Regular polygons
Rotations of varying magnitudes and signs
Rotation symmetry
Shark attack
The side splitting theorem
Similarity
The AA and SAS similarity theorems
The SSS similarity theorem
The side splitting theorem
Size changes
Properties of size changes
Proportions
Similarity
The fundamental theorem of similarity
The transformation Sk
SsA condition and HL congruence
Reflections and minimum distance
Area of tangent circles
Area of tangent circles
Area of tangent circles
Tessellations
Area of trapezoids
Constructing a trapezoid
Find the hidden treasure
Create, measure, and change a triangle
Area of a triangle
Triangle in a rectangle
Triangles in a trapezoid
Translations and vectors
Vertical angles
Creating a toolbar
to reference formulas
p. 115
The following materials were produced by the Glenbrook South High School mathematics department and may not be
reproduced without permission (phone: 847-486-4683 or e-mail: [email protected]).
p. 116
The following materials were produced by the Glenbrook South High School mathematics department and may not be
reproduced without permission (phone: 847-486-4683 or e-mail: [email protected]).