ADVANCED GEOMETRY LAB PREREQUISITE: CREATING CONGRUENT TRIANGLES
Let’s start right away by creating congruent triangles in three different ways: by a translation (a shift),
by a reflection, and by a rotation.
Congruence by Translation
Let’s start by creating a coordinate axis system in Geometer’s Sketchpad. Under the Graph menu on the
top of the screen, choose Show Grid. Next, also under the Graph menu, choose Snap Points. (This
ensures that points are placed on x and y values that are on the grid and hence have precise integer
values). Select the Point tool from the toolbox on the left and plot points at (-3, 4), (-5, 2), and (3, -1).
Use the Text tool to label (-3, 4) as A, to label (-5, 2) as B, and to label (3, -1) as C. Then, connect the
points using the Segment tool. Then, select the Arrow tool and click in open space to reset. Your image
should look as follows:
Let’s shift the triangle 5 units to the right. To do this, we need to set up a vector which defines the
horizontal shift. To do this, let’s create a point exactly 5 units to the right of one of the points on the
triangle (let’s use point C). So, since C is at (3, -1), let’s create another point at (8, -1). Using the Text
tool, let’s label the point as C’. Now click on the Arrow tool and the open space to deselect everything.
Next, we mark a vector by highlighting the point at (3, -1) first and then the point at (8, -1) second.
Under the Transform menu, choose Mark Vector. You’ll see the vector shoot left to right between the
points. Now, highlight all three segments and all three points of the triangle and then from the
Transform menu choose Translate (and then click on the Translate button) in order to shift by the
marked vector. Click in the open space to deselect everything. Then, choose the Text tool and click on
each new point to provide labels.
Now, consider each set of coordinates:
A (-3, 4) and A’ (2, 4)
B (-5, 2) and B’ (0, 2)
C (3, -1) and C’ (8, -1)
Notice, the x-coordinates of all of the images are just 5 greater than the original x-values. (For example,
the x-coordinate of A is -3, while the x-coordinate of A’ is -3+5 = 2.) This should make sense! We had a
horizontal shift of 5 units to the right, hence the x-values increased by 5.
Also note that the y-coordinates are unchanged for each point and its image. (For example, the ycoordinate of A is 4 and the y-coordinate of A’ is also 4). This should make sense! We did not have a
vertical shift at all, so the y-values are unchanged.
Finally, let’s identify all corresponding congruencies:
We summarize by defining the triangle congruence:
Congruence by Reflection:
Go to the File menu to open a New Sketch. Again, under the Graph menu choose Show Grid and then go
back into the Graph menu to select Snap Points. Using the Point tool, plot points at (2, 3) and label it A,
then at (10,2) and label it B, and finally at (10, -3) and label it C. With the segment tool, create the sides
of the triangle. Select the Arrow tool and click in the open space to deselect everything.
Next, our goal is to reflect this triangle over the y-axis. To do this, we simply double-click on the y-axis.
(Recall the y-axis is the vertical axis (up and down!)). The y-axis will “jump out” at you to show you it is
selected. Now, highlight all three segments and all three points of the triangle. Then, from the
Transform menu, choose Reflect. You will see a mirror-image of the original triangle on the opposite
side of the y-axis. Choose the Text tool and click on the new points to label them.
Now, consider each set of coordinates:
A (2, 3) and A’ (-2, 3)
B (10, 2) and B’ (-10, 2)
C (10, -3) and C’ (-10, -3)
Note: The x-coordinate of each image is opposite the original x-value. (For example, the x-coordinate of
point B is 10, while the x-coordinate of point B’ is -10.) This should make sense! The image is on the
opposite side of the y-axis and equidistant from the y-axis, and hence the x-values just have the
opposite sign.
Also note that the y-coordinates are unchanged. This also should make sense! The image does not have
a change in its vertical positioning at all. Hence, the y-coordinates do not change.
Again, we have the six congruencies:
Again, we summarize by defining the triangle congruence:
Congruence by Rotation:
Go to the File menu to open a New Sketch. Again, under the Graph menu choose Show Grid and then go
back into the Graph menu to select Snap Points. Using the Point tool, plot points at (2, 2) and label it A,
then at (5, 3) and label it B, and finally at (4, 6) and label it C. Use the Segment tool to create the sides
of the triangle. Select the Arrow tool and click in the open space to deselect everything.
Next, our goal is to rotate the triangle 90 degrees about the origin (in other words, about the point (0,
0)). To do this, we simply double-click on the origin. The point at the origin will “jump out” at you to
show you it is selected. Now, highlight all three segments and all three points of the triangle. Then,
from the Transform menu, choose Rotate. Type in 90 for the degree of rotation and then click on the
Rotate button. You will see the rotated image. Choose the Text tool and click on the new points to label
them.
To help visualize the 90 degree rotation, note that point A moves along a circular arc through an angle of
90 degrees to form point A’ as shown below.
The same can be shown with the other coordinates. Note here that point B is rotating along an arc by
exactly 90 degrees to form point B’:
Note here that point C is rotating along an arc by exactly 90 degrees to form point C’:
8
C'
4
A'
-15
-10
C
6
B'
B
2
A
-5
5
10
15
-2
-4
-6
-8
Now, consider each set of coordinates:
A (2, 2) and A’ (-2, 2)
B (5, 3) and B’ (-3, 5)
C (4, 6) and C’ (-6, 4)
Note: The x-coordinate of each original point becomes the y-value of each image. (For example, the xcoordinate of point B is 5, while the y-coordinate of point B’ is also 5.) This should make sense! The
entire x-axis just becomes the y-axis when you rotate the x-axis by 90 degrees. Hence, the x-values just
become y-values when you rotate by 90 degrees.
Also note that the x-coordinate of the image is the opposite of the y-coordinate of the original point.
This also should make sense! The upper (positive) part of the y-axis becomes the left (negative) part of
the x-axis when we rotate by 90 degrees. Hence, the y-coordinates of the original points become the
opposite (negative) x-values of the images.
Again, we have the six congruencies and the triangle congruence. Now – on to the lab!
GEOMETER’S SKETCHPAD LAB: STRANGE LITTLE ALIENS CREATING CONGRUENT TRIANGLES
Aliens Phlipp, Tern and Slyde arrive unexpectedly from planet Trans 4M.
Oddly, what these aliens like to do is to transform triangles whenever they see them.
Phlipp has Some Fun: For a geometry project, suppose you create a triangle with coordinates A(5, 3),
B(-2, 6) and C (0, 1). Phlipp comes along and decides to reflect your triangle about the y-axis (creating a
new triangle A’B’C’). Then, Phlipp takes the new triangle A’B’C’ and reflects that about the x-axis to
create triangle A”B”C”.
a)
Create the original triangle in GSP along with the intermediate image (A’B’C’) and final image
(A”B”C”).
b) List the coordinates for the original triangle and for the new points on Phlipp’s intermediate and
final triangles.
A: ( , )
A’: ( , )
A”: ( , )
c)
B: ( , )
B’: ( , )
B”: ( , )
C: ( , )
C’: ( , )
C”: ( , )
Using a complete sentence, explain how the x-coordinates in the intermediate triangle relate to
the x-coordinates in the original triangle.
d) Using a complete sentence, explain how the y-coordinates in the intermediate triangle relate to
the y-coordinates in the original triangle.
e) Using a complete sentence, explain how the x-coordinates in the final triangle relate to the xcoordinates in the intermediate triangle.
f)
Using a complete sentence, explain how the y-coordinates in the final triangle relate to the ycoordinates in the intermediate triangle.
Slyde has Some Fun:
For a geometry project, suppose you create a triangle with coordinates A(7, 5), B(4, -2) and C (-1, 1).
Slyde comes along and decides to translate your image 5 units to the left to form triangle A’B’C’. Slyde
then shifts triangle A’B’C’ down 3 units to create triangle A”B”C”.
a)
Create the original triangle in GSP along with the intermediate image (A’B’C’) and final image
(A”B”C”).
b) List the coordinates for the original triangle and for the new points on Slyde’s intermediate and
final triangles.
c)
A: ( , )
A’: ( , )
A”: ( , )
B: ( , )
B’: ( , )
B”: ( , )
C: ( , )
C’: ( , )
C”: ( , )
Using a complete sentence, explain how the x-coordinates in the intermediate triangle relate to
the x-coordinates in the original triangle.
d) Using a complete sentence, explain how the y-coordinates in the intermediate triangle relate to
the y-coordinates in the original triangle.
e) Using a complete sentence, explain how the x-coordinates in the final triangle relate to the xcoordinates in the intermediate triangle.
f)
Using a complete sentence, explain how the y-coordinates in the final triangle relate to the ycoordinates in the intermediate triangle.
Tern has Some Fun:
For a geometry project, suppose you create a triangle with coordinates A(-4, 0), B(1, 5) and C (3, 2).
Tern comes along and decides to rotate your image 90 degrees about the origin to form triangle A’B’C’.
Then Tern rotates A’B’C’ by 90 degrees about the origin.
a)
Create the original triangle in GSP along with the intermediate image (A’B’C’) and final image
(A”B”C”).
b) List the coordinates for the original triangle and for the new points on Tern’s intermediate and
final triangles.
c)
A: ( , )
A’: ( , )
A”: ( , )
B: ( , )
B’: ( , )
B”: ( , )
C: ( , )
C’: ( , )
C”: ( , )
Using a complete sentence, explain how the x-coordinates in the intermediate triangle relate to
the y-coordinates in the original triangle.
d) Using a complete sentence, explain how the y-coordinates in the intermediate triangle relate to
the x-coordinates in the original triangle.
e) Using a complete sentence, explain how the x-coordinates in the final triangle relate to the ycoordinates in the intermediate triangle.
f)
Using a complete sentence, explain how the y-coordinates in the final triangle relate to the xcoordinates in the intermediate triangle.
Note: The final image is just a 180 degree rotation of the original triangle.
a) Using a complete sentence, explain how the x-coordinates in the final triangle relate to the xcoordinates in the original triangle.
b) Using a complete sentence, explain how the y-coordinates in the final triangle relate to the ycoordinates in the original triangle.
Well, that is all for Phlipp, Slyde and Tern. Despite their annoying tendencies to transform our triangles,
let’s see if maybe they helped us understand these tranformations a little better. Your goal below is to
identify the coordinates of each newly formed triangle without needing Sketchpad to create the image.
1.
An original triangle has coordinates A(4, 4), B(6, 0) and C(-2, -2). Suppose Slyde translates the
triangle 7 units to the right. List the x and y coordinates of the new triangle below:
A’: ( , )
2.
B’: ( , )
C’: ( , )
The same original triangle has coordinates A(4, 4), B(6, 0) and C(-2, -2). Suppose Slyde translates
the triangle 2 units upward. List the x and y coordinates of the new triangle below:
A’: ( , )
B’: ( , )
C’: ( , )
3. An original triangle has coordinates A(1, 5), B(6, 0) and C(-2, 1). Suppose Phlipp reflects the triangle
about the x-axis. List the x and y coordinates of the new triangle below:
A’: ( , )
B’: ( , )
C’: ( , )
4. The same original triangle has coordinates A(1, 5), B(6, 0) and C(-2, 1). Suppose Phlipp reflects the
triangle about the y-axis. List the x and y coordinates of the new triangle below:
A’: ( , )
B’: ( , )
C’: ( , )
5. An original triangle has coordinates A(5, 5), B(6, 2) and C(2, 7). Suppose Tern rotates the triangle
180 degrees about the origin. List the x and y coordinates of the new triangle below:
A’: ( , )
B’: ( , )
C’: ( , )
6. The same original triangle has coordinates A(5, 5), B(6, 2) and C(2, 7). Suppose Tern rotates the
triangle 90 degrees about the origin. List the x and y coordinates of the new triangle below:
A’: ( , )
B’: ( , )
C’: ( , )
Bonus: Suppose Tern rotates this same original triangle (in #5 and #6) 90 degrees clockwise. List the
x and y coordinates of the new triangle: A’: ( , ) B’: ( , )
C’: ( , )