Download Lab 1 Assignment

M 333 L
Spring 2017
Lab Assignment #1
Part I: Geometer’s Sketchpad Tours
Refer to the sections of "Geometer's Sketchpad (Version 4) Learning Guide" which are posted
online in the Handouts folder "Geometer's Sketchpad Learning Guide (Excerpts)". If you purchased
Sketchpad (Version 5), you should still use the posted Version 4 Learning Guide excerpts.

Read pages 9-14

Perform Tour #1 (pp. 16-20)

At the end of Step 15, print out the state of the sketch at that time and hand it in as page 1
of the solutions. Place in a caption the following information:
.
Name, Lab #1, Tour #1, Step 15.

At the end of Step 20, print out the state of the sketch at that time and hand it in as page 2
of the solutions. Place in a caption the following information:
.
Name, Lab #1, Tour #1, Step 20.

NOTE: When printing a sketch, always select “Print Preview” first and then select “Fit to
Page”. Otherwise, in some of the larger sketches, the sketch is split over several pages and
is impossible to see and it wastes paper. Sometimes you need to lower the %-reduction
number at the top a little more for it to reduce to a single page.

Perform Tour #2 (pp. 21-24), print out the final sketch to hand in as page 3 of the solutions.
Place in a caption the following information: Name, Lab #1, Tour #2
Part II: Finding the locus of points which are equidistant to two given points.
1. In a new sketch, draw line segment AB which will remain fixed.
2. Construct a point C and measure the distances CA and CB.
To do this, select the two points and and within the “Measure” menu select “Distance”.
3. Move point C around until the measurements are equal, that is, until CA = CB.
4. Construct 7 more points and measure their distances to A and B respectively and position
them so that each is equidistant to A and B. Position 4 points in one of the half-planes
determined by line AB and position the other 4 points in the other half-plane.
5. Then, complete this conjecture and place it in a caption: "Given a segment, the line which is
perpendicular to the segment and passes through the midpoint of the segment is called the
perpendicular bisector of the segment. The locus of points which are equidistant to two
points A and B is _____________________________________ ."
6. Print the final sketch to hand in as page 4.
Part III: Constructing the Circumcenter and the Circumcircle of a triangle.
1) Construct a triangle, ΔABC . Make the triangle sides be displayed with “Thick” line width.
Measure the length of side AB . To do this, select the segment AB and within the “Measure”
menu select “Distance”. Measure the lengths of the other two sides so that the lengths of all three
sides are displayed.
2) On one of the three sides, construct the perpendicular bisector of that side.
Do this by selecting the side and then within the “Construct” menu select “Midpoint”.
Then, select the midpoint and the side together and then within the “Construct” menu select
“Perpendicular line”.
3) Construct the perpendicular bisectors of a second side of the triangle. Construct the point at
which these two perpendicular bisectors intersect.
There are two ways to construct the point of intersection:
a) Move the Select arrow above the point of intersection until the message in the lower left
hand corner of the window says “Click to construct the intersection” or something similar, and then
click.
b) Select the two lines together and then the “Construct” menu select “Intersection”.
Sometime this method works when method a) doesn’t seem to be available.
4) Construct the perpendicular bisector of the third side of the triangle. Also, make all three
perpendicular bisectors be displayed with “Thin” line width.
Notice how it seems that all three lines are concurrent (that is, all three have a common point of
intersection). This common point of intersection is called the Circumcenter of the triangle.
5) Make the label of the circumcenter point say “Circumcenter”. Do this by selecting the
“Text tool” making the cursor into a hand symbol. Position the hand cursor above the point and the
hand will become black. A single click will make the label disappear (or appear). A double-click
will display the point-properties window. In the point-properties window, type the word
“Circumcenter” in the label field and click on “OK”.
6) Drag the vertices of the triangle around so that the circumcenter is sometimes positioned in
the interior of the triangle, sometimes positioned in the exterior of the triangle, and sometimes
positioned on the triangle.
Can you see what special kinds of triangles have their circumcenters positioned on the triangle?
It’s OK if you don’t at this point.
7) Next, measure  BAC . Do this by selecting points B, A, and C in that order and then
within the “Measure” menu select “Angle”. Then, measure  ABC and measure  ACB .
Now, again drag the vertices of the triangle around as before but pay attention to the measures of
the angles when the circumcenter is positioned on the triangle.
Can you now see what special kinds of triangles have their circumcenters positioned on the
triangle?
8) Next, use the Sketchpad calculator to determine the angle sum of the triangle.
The angle sum of the triangle is the sum of the measures of all three angles of the triangle.
To do this, go to the “Measure” menu and select “Calculate…” . Then, first click on the
measure report for one of the angles, then click on “+” in the calculator. Repeat both clicks for
another angle. Then, click on the measure of the third angle and click on “OK” in the calculaotor.
Drag the vertices around and what how the angle sum changes (or doesn’t change) as the
individual angle measures change.
9) Next, construct a circle which has the circumcenter of the triangle as its center and make the
circle pass through vertex A of the triangle. Make this circle be displayed with “Dashed” line
width.
There are two ways to construct this circle:
a) Select the “Circle Tool” , click and hold on the position of the center point, and then drag
the cursor to the position of a point of the circle (in this case, vertex A) .
b) Select the “Select Arrow Tool”. Select two points, the first is the center and the second is a
point on the radius (here, vertex A). Then, within the “Construct” menu select “Circle by
Center+Point”.
Notice that this circle, centered at the circumcenter and passing through vertex A (or B or C)
passes through all three vertices A, B, and C. Because this circle passes through all vertices of this
polygon, it is said to circumscribe the polygon. Because this circle circumscribes the triangle, it is
called the Circumcircle of the triangle.
Make the label of this circle say “Circumcircle” by selecting the “Text Tool”, moving the hand
cursor above the circle until it becomes black, double-clicking to display the circle-properties
window, typing “Circumcircle”, and hitting “OK”.
10) Next, drag the vertices of the triangle so it is a scalene triangle (no two sides congruent)
such that the circumcenter is in the interior of the triangle.
Change the labels of points A, B, and C to have subscripts, to say A1 , B1 , and C1 ,
respectively. Do this for point A by moving the hand cursor of the “Text Tool” above point A until
it becomes black, double-clicking to display the point-properties window, typing “A[1]”, and
hitting “OK”. Similarly, use “B[1]” for B1 and use “C[1]” for C1 .
Drag the measurements of the sides and of the angles so that they appear below the triangle.
Note: An acute triangle is one in which all three angles are acute angles ( have measure < 90 ).
An obtuse triangle is a triangle in which one of the angles is an obtuse angle ( has measure > 90).
A right triangle is a triangle in which one of the angles is a right angle.
In a caption above the triangle, write the following text filing in the blanks correctly:
“The Circumcenter of a triangle is the point at which the three perpendicular bisectors of the
sides of the triangle intersect.
The triangles that have the circumcenter in the interior are ____________________ triangles.
The triangles that have the circumcenter in the exterior are ____________________ triangles.
The triangles that have the circumcenter on the triangle are ____________________ triangles.”
Place the caption above the triangle. Position the labels so that they are clearly visible. Then, print
the sketch (Use “Fit to Page” within “File / Print Preview” first) and hand it in as page 5.
End of Lab #1 Assignment
New terms defined in this assignment:
Circumcenter of a triangle,
Circumcircle of a triangle,
to circumscribe a triangle or other polygon,
Scalene triangle, Obtuse triangle, Acute triangle,
Important Sketchpad processes learned in this lab:
Measuring the length of a segment,
Measuring an angle,
Constructing the midpoint of a segment,
Constructing the perpendicular bisector of a segment,
Constructing the point of intersection of geometric objects,
Altering the label of a point, of a circle, or of any other constructed object,
Making a label which has a subscript, as in A1 , B1 , or C1 .
Using the Sketchpad calculator to make calculations involving measurements made earlier.
You will need to perform the above processes in the future. Review this assignment periodically to
recall how to do these processes within Geometer’s Sketchpad.
As always when a Sketchpad sketch is assigned, print out the sketch to be turned-in.
If you don't have Sketchpad at home, you should work the assignment in the Math Computer Lab in
the RLM building in RLM 7.122. There is a copy of the Learning Guide and the User's Manual in
an envelope along the back wall of the lab. For information on using it there, read the handout
"Using Geometer's Sketchpad in the Math Lab" available through a link on the class webpage.