ADVANCED GEOMETRY: SIMILARITY LAB
Our goal today is to use “indirect” reasoning to determine an unknown length. This means that we will
determine a measurement without a direct measurement (and instead using logic, algebra and
geometric relationships). After working through the Sketchpad task, some of us may have time to try a
physical experiment.
So, here we go!
Let’s imagine that we want to determine the length of a flag pole. Of course, this would be difficult to
measure directly (with a tape measure). So, let’s determine the length of the pole using indirect
measurement (i.e. mathematical reasoning).
1. Open a new Sketchpad File. Start by creating a horizontal line (not segment) along the bottom
of a new sketch. Label the control points as A and B. But then, hide the control points.
2. Create two new points on the horizontal line (point C somewhere on the left hand side of the
line, and point D somewhere on the right hand side of the line.).
3. Construct a perpendicular to the original line AB through point D. Plot a point near the top of
the perpendicular line and label it as E. (That is the top of the flag pole). Now, hide the
perpendicular line and then create a segment from D to E. Segment DE is your flag pole.
Note: Certainly we could use GSP to measure DE right now, but let’s not do that. Remember,
we are imagining that we cannot directly measure this length! Not easy to measure the length
of a flagpole!
4. Construct a perpendicular to the original line AB through point C. Now, let’s imagine a person
whose eyes are roughly 5 feet above the ground. To create a “5 foot” distance, highlight point C
and then choose translate from the transform menu. When the pop-up appears, change the
fixed distance 5 cm. (We’ll let each cm represent 1 foot). Leave the angle at 90 degrees. This
will create a point 5 cm above the line along the perpendicular (90 degrees). Label the point as
F. So, we are thinking of segment CF as the distance from the ground to the eye of a person,
and treating this length as 5 feet instead of 5 cm. Hide line CF and replace with segment CF.
5. Since point F is the eye of the person, let’s create a small circle centered at F to represent the
head of the person. When you create this circle, attach the control point to segment CF.
(Choose a reasonable size for the “head”!) Your sketch should look roughly as follows:
E
F
C
6.
D
Now, let’s place a mirror on the ground between you (CF) and the flagpole (DE). To do this,
“lock” a small segment onto line CD between points C and D. Immediately after creating the
segment (while it is still highlighted), change the line width to thick using the Display menu.
Label the endpoints as M and R (for mirror). Next, hide line CD. You should be able to see your
mirror lying on the ground between you and the pole. See below:
E
F
C
M
R
D
7. Now, let’s create a ray of light from the top of the flagpole to the mirror. To do this, choose the
ray tool and construct a ray from E to a point near the middle of the mirror. Lock the control
point of the ray onto the mirror. And, change the line width of the ray back to thin (using the
display menu). Label the new point on the mirror as P. See below:
E
F
D
M P R
C
8. Now, let’s reflect the light ray off of the mirror. To do this, select the mirror as the “mirror” by
double clicking on the mirror. You should see the segment representing the mirror “jump out”
at you. Now, select the light ray and reflect it about the mirror using the transform menu. The
image should look roughly as follows:
E
F
C
M
P
R
D
9.
Now, move point C to move the person toward the mirror until the eye makes contact with the
reflected light ray. So, now the person can see the top of the flag pole in the mirror. The
diagram should look roughly as follows:
E
F
C
M
P
R
D
10. Now, make an argument that triangle FCP is similar to triangle EDP. (See lab sheet.)
11. Certainly, several measurements can be taken directly using a tape measure (in real life). Which
three measurements can be taken that might help us be able to solve for the height of the
flagpole? (Hint: use sides of the similar triangles).
12. Now, use GSP to find the measurements that you defined above. Write the measurements on
the diagram provided to you. Remember to label the measurements as feet instead of cm, since
we are letting each cm in GSP represent 1 foot in real life.
13. Showing your work in the space provided on the lab sheet, solve for the height of the flag pole.
Remember to label your answer in feet.
14. Now, let’s see how close we came. Use GSP to measure the height of the flag pole. Our
mathematical calculation may not match exactly, but will hopefully be close.
15. Identify at least one source of error that is not allowing us to get matching results.
16. Now, for a bonus if time allows, see if you can use this technique to determine the height of our
ceiling without actually measuring that height. Show a diagram and all calculations below.
GSP SIMILARITY LAB:
As you work through the lab on the computer, there are a few problems that require your response.
Use this sheet to respond to those questions.
10. Make an argument that triangle FCP is similar to triangle EPD.
E
F
C
M
P
R
D
11. Certainly, several measurements can be taken directly using a tape measure (in real life). Which
three measurements can be taken that might help us be able to solve for the height of the
flagpole? (Hint: use sides of the similar triangles).
12. Now, use GSP to find the measurements that you defined above. Write the measurements on
the diagram provided to you. Remember to label the measurements as feet instead of cm, since
we are letting each cm in GSP represent 1 foot in real life.
13. Showing your work, solve for the height of the flag pole. Remember to label your answer in
feet.
14. Now, let’s see how close we came. Use GSP to measure the height of the flag pole. Our
mathematical calculation may not match exactly, but will hopefully be close.
15. Identify at least one source of error that is not allowing us to get the precise result.
16. Now, for a bonus if time allows, see if you can use this technique to determine the height of our
ceiling without actually measuring that height. Show a diagram and all calculations below.