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Grade 9 Mathematics
Geometer Adventure
Table of Contents
1. Construct, Don’t Draw ..................................................................2
2. Geometer Sketchpad: Helpful Hints ............................................3
3. The Sum of the Angles in a Triangle (SATT)...............................4
4. The Triangle Inequality .................................................................5
5. Exterior Angles ..............................................................................6
6. Interior Angles ...............................................................................8
7. Pythagorean Theorem ....................................................................9
8. Medians of a Triangle ..................................................................10
9. Altitudes of a Triangle .................................................................10
10. Perpendicular Bisectors of a Triangle .........................................10
11. Angle Bisectors of a Triangle ......................................................10
12. Angles in a Circle ........................................................................11
13. Midpoint and Diagonal Conjectures ...........................................13
14. Reflections in the Coordinate Plane ............................................14
15. Translations in the Coordinate Plane ..........................................15
16. Glide Reflections .........................................................................16
17. Symmetry .....................................................................................18
18. Rotation Tesselations...................................................................20
Grade 9 Mathematics
Geometer Adventure
1. Construct, Don’t Draw
The humble right triangle
A
Use the line segment tool, and draw a right angled triangle.
A
When finished, use the select tool to drag point A to the left. Triangle ABC is
no longer a right angled triangle.
C
Start a new sketch. Draw a vertical line segment. Select the line
segment and a point B which is on the line segment, and construct a perpendicular line. (Construct 
Perpendicular Line)
B
Geometer will construct a line perpendicular to the selected line and through the selected point.
Create a point A, on the vertical line.
Draw a line segment from point A to the line forming a
right angled triangle. Call the last vertex of the triangle
point C.
Try dragging any of the points A, B or C to distort the
right angled triangle.
ABC will always be a right triangle!!!
Trick: select the line through BC and hide it (Display menu or use Command H); then draw a line segment
through BC creating a triangle guaranteed to be right.
Activities
Construct an equilateral triangle using transformations.
Draw any line segment AB. Select point B, and under the Transform menu,
mark centre “B”. This marks point B as the centre of rotation. Select the line
segment and point A, and using the Transform menu, rotate by 60 degrees.
(counterclockwise is positive 60º, clockwise is -60º)
Try to complete the equilateral triangle by marking a point as centre, then
rotating a point and line by 60º.
Drag any of the three points to distort ABC, but it will always be and equilateral triangle.
Using Geometer, construct a square, rectangle, pentagon, hexagon, and octagon. Make the pentagon, hexagon
and octagon regular, that is all sides equal and all angles equal.
Grade 9 Mathematics
Geometer Adventure
2. Geometer Sketchpad: Helpful Hints
Undo
The undo command on the Edit menu undoes the last step, all the way back to the
beginning of the Sketchpad document.
Redo
Under the Edit menu, you can redo a command if you undo one step too many.
Multiple Selections Select several object at a time by clicking on them.
hot-keys
Many features and commands are used repeatedly and have shortcut keystrokes. For
example, click on the File menu and read the hot-keys to the right of each command.
Preferences
Explore the preferences that you can set automatically under the Edit menu.
Hidden Tools
Hold down the selection tool button (arrow); there are rotate and dilate tools.
Hold down the line segment tool
, there are ray
and line tools
.
Constructing Lines and Segments
Place two points anywhere on the page. Select both points and click the Construct menu…
If line segment is selected, the Construct menu will create a line segment.
If ray is selected, the Construct menu will create a ray (select the points in order).
If line is selected, the Construct menu will create a line.
Explore some features
1.
Why are some commands grey and some are black?
2. How do you know that an object has been selected?
3. How do you turn the label of an object on and off? How do you rename a label? How do you move labels?
4. Using the selection tool, double click on a point. What does this do?
5. Using the selection tool, double click on a line. What does this do? Does it matter if you double click on a line
segment or a ray?
6. How do you change the font and size of the labels and text?
7. How do you measure an angle?
8. Does Sketchpad measure reflex angles?
Grade 9 Mathematics
Geometer Adventure
3. The Sum of the Angles in a Triangle (SATT)
Geometer Tidbits:
If a Measure menu feature is greyed out (not available), check if you have selected the appropriate
ingredients. For example, you cannot measure an angle unless exactly 3 points are selected.
Investigation:

Construct any triangle and label the vertices A, B and C

Select vertices A, B and C in that order

Now measure the angle ABC (using the Measure menu). In the
same way measure angle BCA and angle CAB.

Use the calculator (Measure  Calculate…) to find the sum of
the three angles. Do not type in the actual measurements;
simply highlight each of the three angle measurements with
plus signs in between.

Making a table:
m ABC = 29°
m BCA = 80°
A
m CAB = 71°
m ABC + m BCA + m CAB = 180°
B
C
Select all three angle measurements and the angle sum.
Construct the table by selecting tabulate from the Graph menu.

Adding to your table of values:
Change the current triangle by dragging a side or vertex to a new position.
Double-click on the table using the pointer tool. The new values should be displayed as a new column in the
table.
Make four more entries in your table. (Note that you can move the entire table to a different location in your
sketch by clicking and dragging.)
Your table might look something like this one:
m ABC
m B CA
m CA B
m ABC+ mCAB+ mBCA
117.07 
27.78 
35.14 
180.00 
84.74 
47.39 
47.88 
180.00 
75.96 
70.12 
33.92 
180.00 
79.28 
49.73 
50.99 
180.00 
98.23 
48.93 
32.84 
180.00 
When you change the shape of the triangle what happens to the angle measurements and what happens to the
angle sum?
Grade 9 Mathematics
Geometer Adventure
4. The Triangle Inequality

Open a new sketch and construct a triangle.

Label the vertices of the triangle
. (You can move the vertex labels so that they are not on top of the
vertices - use the finger tool plus click and hold.)
If the vertices are not labeled A, B and C, relabel them by double-clicking the finger tool. This
will allow a label change.
A

B
Select side AB.
C
Using the Measure menu, determine the length of side AB. Now measure the length of side AC.
Select the measurements of the lengths of AB and AC.
Use the calculator (Measure menu) to find the sum of the lengths of sides AB and AC.

Now measure the length of side BC. Compare the sum of the lengths of AB and AC with the length of BC.

Create a table as you did in Lab 3 by selecting tabulate from the Graph menu.

Use the pointer to change the shape of your triangle by dragging a vertex to a new location. Again, compare
the sum of the lengths of AB and AC to the length of BC.
Repeat this for several different triangles which are produced by dragging a vertex or a side.
Your table may look something like this one:

m AB+ m CA
m BC
18.34 cm
7.94 cm
18.27 cm
9.28 cm
16.76 cm
9.27 cm
12.01 cm
7.86 cm
13.50 cm
6.77 cm
12.08 cm
6.77 cm
12.08 cm
6.77 cm
Use one of the symbols > , < or = to complete the relation:
Restate the relation in words by completing the sentence:
The sum of the lengths of two sides of a triangle is ____________________ the third side.
This relationship is the triangle inequality.

Move vertex A closer to side BC.

How does this affect the comparison between the sum of the lengths of two sides and the length of the third
side?

What will happen to the triangle inequality if you move vertex A onto side BC?

In your opinion, if vertex A lies on segment BC is the figure ABC still a triangle?
Grade 9 Mathematics
Geometer Adventure
5. Exterior Angles
Exterior Angle of a Triangle

A
C
D
Construct ∆ABC where the base lies on a ray AC as shown.
Vertices A and C should be the control points of the ray.
Construct a point D on the ray AC.
B
The angle BCD is an exterior angle of the ∆ABC. Measure angle BCD.
Angles CAB and ABC are in the interior of the triangle and are on the opposite side of the triangle from angle
BCD. These angles are called interior opposite angles.
Measure angle CAB and angle ABC then use the calculator to find their sum.

Make a GSP table showing the value of angle BCD, the values of angles CAB and ABC as well as the sum of
these interior angles.

Now add four more different entries to your table by dragging the triangle so that the angles change and
double-clicking on the table.

What do you notice about the relation between the exterior angle and the sum of the two interior opposite
angles?
Sum of the Exterior Angles of a Triangle

Use three rays to construct a triangle as shown. (Remember to use the control
points as vertices.)

Measure all three exterior angles and find their sum. Create a GSP table using
values for several different triangles.

What do you notice about the sum of the exterior angle of a triangle?
Sum of the Exterior Angles of a Quadrilateral

Now construct any quadrilateral using four rays.

Measure the four exterior angles and find their sum.

Create a table using results for several different quadrilaterals. What is your
conclusion?
Grade 9 Mathematics
Geometer Adventure
Sum of the Exterior Angles of a Polygon

Use Sketchpad to find the sum of the exterior angles of a pentagon.
A conjecture is a sort of educated guess about something. What is your conjecture about the sum of the
exterior angles of any polygon?

Test your conjecture for a dodecagon. (A dodecagon is a polygon having 12 sides.) Can you think of a way to
prove your conjecture for all polygons?

Recreate a sketch for the exterior angles of a heptagon.
Measure the exterior angles and their sum as before. Now
drag one of the vertices toward the centre of the heptagon
until it looks like the one below. (This heptagon which goes in
on itself is called a concave heptagon.)
What has happened to the sum of the exterior angles?
Can you think of an explanation for this?
(Hint: It has to do with the way Sketchpad measures angles)
6.
Grade 9 Mathematics
Geometer Adventure
6. Interior Angles
Interior Angles of a Quadrilateral

Now construct a quadrilateral. Because of Sketchpad's inability to deal with reflex angles, do not construct a
concave quadrilateral.

Measure the four interior angles and find their sum.

Create a GSP table using results for several different quadrilaterals. What is your conclusion?
Interior Angles of Polygons

Create a pentagon, making sure that it is not concave.

Measure the interior angles and find their sum.

Drag one or more of the vertices.

What do you notice about the sum of the interior angles of a pentagon?

Repeat this for different types of polygons and complete the following chart.
POLYGON
NUMBER OF
SIDES
SUM OF THE
INTERIOR
ANGLES
SUM OF THE
EXTERIOR
ANGLES
Triangle
3
180°
360°
Quadrilateral
4
360°
360°
Pentagon
6
7
Octagon
Nonagon
10
100-gon
n-gon
100
n
Grade 9 Mathematics
Geometer Adventure
7. Pythagorean Theorem
Verify the Pythagorean Theorem
The sum of the squares of a right angled triangle is equal to the sum of the squares of the other two sides;
or, ∆ABC is a right triangle, where  C = 90º, if and only if a 2  b 2  c 2 .

construct a right angled triangle labeled ABC, where C is the right angle

label the sides a, b and c

construct squares from each side by rotating line segments (remember positive angles are counter clockwise,
negative angles are clockwise)

select the 4 points surrounding each square, and construct a quadrilateral interior; repeat for the other 2
squares

select the area, and measure the area of the square; repeat for the other 2 squares

right click on the largest area measurement and select Properties… Label the measurement c^2.

Repeat for the other two area measurements, labeling them a^2 and b^2.

Using the calculate feature, demonstrate the Pythagorean Theorem, by showing that a^2 + b^2 = c^2

colour the squares (optional)
The Converse of the Pythagorean Theorem
Show that the Pythagorean Theorem only works for right angled triangles.

construct an arbitrary triangle ABC

using rotated line segments, construct a square at each side of the triangle

select the 4 points surrounding each square, and construct a quadrilateral interior; repeat for the other 2
squares

determine the areas of each of
the squares

label the measurements of the
areas of the squares a^2, b^2
and c^2

measure  ACB

Move any of the points A, B or
C and verify that a 2  b 2  c 2
will be true only if  C = 90º.
c^2
B
c
Area a^2 = 18.12 cm 2
A
a
b
Area b^2 = 34.23 cm 2
(Area a^2 ) + (Area b^2 ) = 52.35 cm 2
Area c^2 = 52.35 cm 2
b^2
C
a^2
Grade 9 Mathematics
Geometer Adventure
8. Medians of a Triangle

A median of a triangle is the line segment that joins a vertex to the midpoint of the opposite side.

Demonstrate that the three medians of a triangle intersect at a single point called the centroid.

The centroid divides each median into two segments. Compare the lengths of the two segments:

Drag one or more of the vertices of the triangle. What do you notice? Complete the following statement:
The centroid divides the median into two line segments. The ratio of the longer segment to the ratio of the
shorter segment is always ____________________.
9. Altitudes of a Triangle
The altitude of a triangle is the perpendicular distance from a vertex to the opposite side. When used to
find the area of a triangle, we call the altitude the "height" of a triangle.
The altitudes of a triangle intersect at the orthocentre.
Under what conditions will the orthocenter be inside the triangle? Outside the triangle? On a side of the
triangle?
10. Perpendicular Bisectors of a Triangle
The perpendicular bisector of a line divides the line in two equal segments and intersects the line at a right
angle.

Verify that all three perpendicular bisectors of a triangle intersect at a common point. Label the point G.

Construct a circle with centre G that passes through each vertex of the triangle. This circle is called the
circumcircle of the triangle. The centre G is called the circumcentre.

Drag one or more of the vertices of the triangle. What do you notice?
11. Angle Bisectors of a Triangle
An angle bisector is a line segment that divides an angle into two equal parts.

Bisect the angles of the triangle. Verify that the angle bisectors will intersect at a point within the triangle.
This point is called the incentre of the triangle. Label this point D.

Hide the angle bisectors.

Construct lines from D to meet each side at 90°. Mark the points
where these lines segments meet each of the sides and label them E,
F and G.

Confirm that DE = DF = DG.

Using D as a centre and DE as a radius, draw a circle. This circle is
called the incircle of triangle ABC.
B
E
G
D
C
F

Under the Display menu, select Show all hidden.

Hide the perpendicular line segments.

Drag one or more of the vertices of the triangle. What do you notice about the circle?
A
Grade 9 Mathematics
Geometer Adventure
12. Angles in a Circle
Inscribed Angles in a Circle
Here are some definitions you should know before beginning this lab:
A chord is any line segment that has its endpoints on the circumference of a circle.
A diameter is a chord that passes through the center of a circle.
An inscribed angle of a circle is an angle that has all 3 of its points on the circumference of a circle. The
chord formed by the two points of the angle that are not the vertex is the
chord that is subtended by the inscribed angle.
Start a new sketch and draw a circle. Construct three points on the circle. (You
can use the control point on the circumference as one of the points if you like.)
Connect the three points with line segments as shown.
The angle DEF is called an inscribed angle because it has all three of its points
on the circumference of a circle.
The angle DEF is the angle subtended at the circumference by the chord DF.
Measure the angle DEF. (Use "Preferences" in the Display menu to set the
precision of angle measurement to 'units'.)

Drag the point E to different positions on the circle staying above chord DF.
What happens to the value of the angle subtended at the circumference?
Create two other angles subtended by the chord DF, on the same side of DF as
point E. Measure the new angles to verify your results.

What happens to the angle subtended at the circumference by the segment DF
when angle DEF is below DF?
Inscribed Angles subtended by a diameter

To construct a diameter, first, draw a line segment GH using the line
segment tool.

Construct the midpoint of GH in order to determine the center of the circle
A.

Using the compass tool, create a circle using center A and drag the compass
tool to make point H land on the circumference of the circle.

Create a point J on the arc of the circle, and measure angle GJH. What do
you notice about angles subtended by a diameter? Move point J around the
arc of the circle to confirm your results.
Grade 9 Mathematics
Geometer Adventure
Sector Angles in a Circle


A sector angle is an angle that has its vertex at the centre of the circle, and the other two points are on the
circumference.
D

Start a new circle sketch like the one shown here, where angle ADB is subtended
by the chord AB, and angle ACB is a sector angle subtended by AB.

Measure angle ADB and angle ACB.

From the Measure menu select Calculate…

Select the measured angle ADB, click on the division symbol, and then click on
the measured angle ACB. What result do you get?

Now drag point D to different positions and observe the measurements of angle
A
ADB and angle ACB. What do you notice?
C
Cyclic Quadrilaterals
A cyclic quadrilateral is a quadrilateral formed by having all 4 points on the circumference of a circle.
Demonstrate the relationships between the angles of a cyclic quadrilateral.
B
Grade 9 Mathematics
Geometer Adventure
13. Midpoint and Diagonal Conjectures
When the midpoints of the sides of a polygon are
joined, a new polygon with the same number of sides
as the original is created. This new polygon is an
inscribed polygon. The segment that joins the
midpoints of two adjacent sides is a midsegment of
the polygon.
A diagonal is any line joining two vertices that are not
adjacent to one another.
Construct a drawing for each of the following
conjectures. Show measurements and calculations to
demonstrate the results of the conjecture.
B
L
C
K
G
A
midsegment
D
J
H
F
I
E
Midsegment Conjectures

Determine the relationships between the midsegment of a triangle and the side opposite the
midsegment.

Determine the shape formed by the midsegments of a quadrilateral.

Determine the shape formed by the midsegments of a rectangle.

Determine the shape formed by the midsegments of a square.

Determine the shape formed by the midsegments of a kite.
(i.e In each case, you should compare the angles created between the midsegments and their lengths)
Diagonal Conjectures

Determine the properties of the diagonals of a parallelogram.

Determine the properties of the diagonals of a rectangle.

Determine the properties of the diagonals of a rhombus

Determine the properties of the diagonals of a square.

Determine the properties of the diagonals of a kite.
(i.e. In each case, you should compare the lengths of the diagonals and how they intersect each other).
You may add any other properties that you discover.
Grade 9 Mathematics
Geometer Adventure
14. Reflections in the Coordinate Plane
Grade 9 Mathematics
Geometer Adventure
15. Translations in the Coordinate Plane
Grade 9 Mathematics
16. Glide Reflections
Geometer Adventure
Grade 9 Mathematics
Geometer Adventure
Grade 9 Mathematics
17. Symmetry
Geometer Adventure
Grade 9 Mathematics
Geometer Adventure
Grade 9 Mathematics
Geometer Adventure
18. Rotation Tesselations
This activity can be done with any figure that tessellates. However, you must keep in mind that the size of the
angle you rotate by must divide evenly into 360º. [This assumes that you rotate by the number of degrees in the
angle you are rotating around.] Note: A quicker way to rotate an item is to hold down the pointer tool and then
select the rotate tool (3rd one in from the left).
i.e. The example here begins with an equilateral triangle with a 60º degree rotation of the figure. A square would
rotate 90º, and a hexagon would need a 120º rotation.
This activity was adapted for use with The Geometer’s Sketchpad - from Geometry (1994 Teacher's Edition),
Jurgensen, Brown, & Jurgensen, by Houghton Mifflin Company.
Activity: A Rotation Tessellation
Construct a line segment AB. Mark point B as the centre of rotation
(double click on B).
D
A
3. Alter the shape of side AB by placing and connecting a series of points
close to, but not on side AB. (Shown at the right) Delete the original side
AB.
C
2. Select the line segment AB and point A and rotate by 60º around point B.
4. Select all of the line segments and points connecting A and B and rotate
them about point B by 60º. Delete the originally rotated line segment AA'.
5. Connect the points A and A 'with a straight line segment.
6. Select the entire drawing and rotate 60º about the point B.
7. Continue to rotate the entire drawing, 60º about the point B, until you get back to the original drawing.
8. Try dragging one of the points you created around side AB. What happens?
*** YOU CAN ALTER THE ORIGINAL FIGURE AT
ANY TIME ***
NOTE: Only 1 point was moved to change the figure
on the left into the figure on the right.
A'
1.
B
Constructing the Basic Shape
Grade 9 Mathematics
Geometer Adventure
A Construction For Those Who Are Feeling Brave!
1.
Follow steps 1 through 5 as in the above directions.
2. Locate the midpoint of side AA'.
3. Alter the left half of side AA'.
4. Rotate the shape drawn in step 3 through 180º about the midpoint of side AA'.
5. Rotate the entire new shape 60º about the point B.
Rotation Tessellation Questions:
1.
Why did the directions specify that you rotate around point B if the triangle was equilateral/equiangular?
2. Try rotating about a different vertex to verify your conjecture.
3. You were directed to rotate using an angle of 60°. Would other angles work also? Why or why not?
4. Investigate using different angles of rotation.
5. Select 1 point and move it. Notice that ALL 6 of your transformed images react. WHY?