Similarity Part B: Similar Triangles (35 minutes)

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

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.
  

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.

Video Segment In today’s professional world, there are many practical applications that rely on triangle similarity. One such application is in medicine, where similar triangles are used to calculate the position of radiation treatment for cancer patients. In this segment, Sandra John-Baptiste and Jason Talkington work with dosimetrist Max Buscher to calculate the position of radiation beams so as to avoid their overlap on the spinal cord. Can you think of any other applications of triangle similarity? You can find the first of these segments on the session video approximately 21 minutes and 10 seconds after the Annenberg Media logo. The second part of this segment begins approximately 22 minutes and 23 seconds after the Annenberg Media logo.

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.

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