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Geometry Session 8: Solutions
Session 8 Part A Part B Part C Homework
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Session 8 Materials:

A B C 


Solutions for Session 8, Part B

See solutions for Problems: B1 | B2 | B3

Problem B1

The teacher places the mirror at point C, a distance ds 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 dt. The teacher knows her height, ht, 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 dt and ds, by similarity of the two triangles, she can say that ht/dt = hs/ds or hs = (ht • ds)/dt. 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.

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Problem B2

We can conjecture that two triangles are similar if two of their respective angles have the same measure.





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Problem B3


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).


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.


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.

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