Learning Math: Geometry
Similarity Part B: Similar Triangles (35 minutes)
Session 8, Part B
In this part
- The Mirror Trick
- Similarity Tests
- Measuring with Shadows
A mathematics teacher likes to astound her students with tricks that can be explained through mathematics. Before the class studies similarity, the teacher brings a mirror to class and performs this trick: Note 3
The teacher puts the mirror on the floor facing up and asks a student to stand two feet from it. The teacher then positions herself so that she can just see the top of the student’s head when she looks in the mirror. With a quick calculation, she reports the student’s height. She’s able to do the trick on every student in class.
It’s important to know how mirrors work to understand this situation: The angle of incidence (the angle at which the light strikes the mirror) is equal to the angle of reflection (the angle at which the light leaves the mirror). Note 5
Problem B1
Sketch the teacher, student, and mirror. Find a pair of similar triangles in your sketch, and explain how the teacher does her mirror trick. Note 4
To further explore similar triangles, go to Learning Math: Measurement, Session 5
Similarity Tests
Problem B2
For each pair of angles given below, sketch at least three different triangles that have these two angle measures. What do you notice? Note 6
a. | 90° and 60° |
b. | 45° and 45° |
c. | 120° and 30° |
d. | 80° and 40° |
In Part A, we outlined an argument that if two triangles have all three pairs of sides in proportion, the triangles must be similar. This is the SSS similarity test.
There are two other similarity tests:
- AAA similarity: If two triangles have corresponding angles that are congruent, then the triangles are similar. Because the sum of the angles in a triangle must be 180°, you really only need to know that two pairs of corresponding angles are congruent to know the triangles are similar.
- SAS similarity: If two triangles have two pairs of sides that are proportional and the included angles are congruent, then the triangles are similar.
Measuring with Shadows
Using shadows is a quick way to estimate the heights of trees, flagpoles, buildings, and other tall objects. To begin, pick an object whose height may be impractical to measure, and then measure the length of the shadow your object casts. Also measure the shadow cast at the same time of day by a yardstick (or some other object of known height) standing straight up on the ground. Since you know the lengths of the two shadows and the length of the yardstick, you can use the fact that the sun’s rays are approximately parallel to set up a proportion with similar triangles.
Because the sun’s rays are parallel, the triangles are similar. Thus:
Unknown Height = Known Height
Shadow a Shadow b
Problem B3
On a sunny day, Michelle and Nancy noticed that their shadows were different lengths. Nancy measured Michelle’s shadow and found that it was 96 inches long. Michelle then measured Nancy’s shadow and found that it was 102 inches long.
a. | Who do you think is taller, Nancy or Michelle? Why? |
b. | If Michelle is 5 feet 4 inches tall, how tall is Nancy? |
c. | If Nancy is 5 feet 4 inches tall, how tall is Michelle |
Notes
Note 3
If you are working in a group, you may want to start by actually doing the “mirror trick” described here. When you calculate, remember that you should use a number closer to the height of your eyes rather than your actual height, but position yourself so that you just see the top of the other person’s head. For Problem B1, you can ask others to figure out your height based on the setup used and the other person’s (now known) height.
Note 4
If you are working in a group, compare the triangles you made with the triangles other people made and discuss what you notice.
Note 5
The mirror acts like there’s an exact copy of the world directly behind it that you are peeking into. If you connect your eyes to the top of the other person’s head, as if the line went through the mirror, the angles would be the same because they are vertical angles. So the reflected angles are the same as the “behind-the-mirror angles,” which are the same as the angles at which you are looking.
Note 6
If you’re working in a group, have each participant create just one triangle for each part, and then compare the results as a group.
Solutions
Problem B1
The teacher places the mirror at point C, a distance d_{s} away from the student (see picture). She then steps away from the mirror until she sees the top of the student’s head in the mirror. Let’s call the distance from the teacher to the mirror d_{t}. The teacher knows her height, h_{t}, and she knows that the angle of incidence equals the angle of reflection when a beam of light hits a reflective surface. We call this angle ß. Since the triangles ABC and DEC are right triangles and since they share the angle ß, they are similar. So the teacher knows that, once she measures d_{t} and d_{s}, by similarity of the two triangles, she can say that h_{t}/d_{t} = h_{s}/d_{s} or h_{s} = (h_{t} • d_{s})/d_{t}. In other words, by knowing her own height, and by measuring her own as well as the student’s distance from the mirror, she can calculate the student’s height.
Problem B2
We can conjecture that two triangles are similar if two of their respective angles have the same measure.
a. | |
b. | |
c. | |
d. |
Problem B3
a. | Nancy is taller. Since the right triangles defined by their heights and their shadows are similar, then the bases of the triangles have to be proportional to the heights of the triangles (i.e., their body heights). |
b. | Converting Michelle’s height into inches (64 inches) and setting up a proportion, you would have: 64 / x = 96 / 102, or x = 68 Converting 68 inches back to feet, Nancy is 5 feet 8 inches tall. |
c. | Converting Nancy’s height into inches (64 inches) and setting up a proportion, you would have: 64 / x = 102 / 96, or x = 60.24 Converting 60.24 inches back to feet, Michelle is approximately 5 feet and 1/4 inch tall. |