Math 8 Module 3 End of Module Study Guide

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1 Name ANSWER KEY Date 3/21/14 Lesson 8: Similarity 1. In the picture below, we have a triangle DEF that has been dilated from center O, by scale factor r = ½. The dilated triangle is noted by D E F. We also have a triangle D EF, which is congruent to triangle DEF (i.e., ΔDEF ΔD EF). Describe the sequence of a dilation followed by a congruence (of one or more rigid motions) that would map triangle D E F onto triangle D EF. Triangle D'E'F' will need to be dilated from center O, by scale factor r = 2 to bring it to the same size as triangle DEF. This will produce the triangle noted by DEF. Next, triangle DEF will need to be reflected across line EF. The dilation followed by the reflection will map triangle D'E'F' onto triangle D''EF.

2 Lesson 9: Basic Properties of Similarity Use the diagram below to answer questions 2 and Which triangles, if any, have similarity that is symmetric? S ~ R and R ~ S. S ~ T and T ~ S. T ~ R and R ~ T. 3. Which triangles, if any, have similarity that is transitive? One possible solution: Since S ~ R and R~ T, then S~ T. Note that U and V are not similar to each other or any other triangles. Therefore, they should not be in any solution.

3 Lesson 10: Informal Proof of AA Criterion for Similarity 4. Are the triangles shown below similar? Present an informal argument as to why they are or why they are not. Yes, ABC~ A'B'C'. They are similar because they have two pairs of corresponding angles that are equal. Namely, B = B' =45, and A = A' =45

4 5. Are the triangles shown below similar? Present an informal argument as to why they are or why they are not. No, ABC is not similar to A'B'C'. They are not similar because they do not have two pairs of corresponding angles that are equal. Namely, A A' and B B'.

5 Lesson 11: More about Similar Triangles 6. In the diagram below, you have ABC and A B C. Based on the information given, is ABC~ A B C? Explain. Since there is only information about one pair of corresponding angles, we need to check to see if corresponding sides have equal ratios. That is, does AB / A B' = AC / A'C', or does (3.5)/ (8.75) =6/21? The products are not equal: Since the corresponding sides do not have equal ratios, the triangles are not similar.

6 7. In the diagram below, ABC~ DEF. Use the information to answer parts (a) (b). a. Determine the length of AB. Show work that leads to your answer. Let x represent the length of AB. Then x = We are looking for the value of x that makes the fractions equivalent. Therefore, = 13.23x and x = 8.4. The length of AB is 8.4 b. Determine the length of DF. Show work that leads to your answer. Let y represent the length of DF. Then 4.1 = 6.3. We are looking for the value of y that makes the y fractions equivalent. Therefore, = 6. 3y and = y. The length of DF is

7 Lesson 12: Modeling Using Similarity 8. Henry thinks he can figure out how high his kite is while flying it in the park. First, he lets out 150 feet of string and ties the string to a rock on the ground. Then he moves from the rock until the string touches the top of his head. He stands up straight, forming a right angle with the ground. He wants to find out the distance from the ground to his kite. He draws the following diagram to illustrate what he has done. a. Has Henry done enough work so far to use similar triangles to help measure the height of the kite? Explain. Yes, based on the sketch, Henry found a center of dilation, point A. Henry has marked points so that when connected would make parallel lines. So the triangles are similar by the AA criterion. Corresponding angles of parallel lines are equal in measure, and BAC is equal to itself. Since there are two pairs of corresponding angles that are equal, then BAC~ DAE.

8 b. Henry knows he is 5 ½ feet tall. Henry measures the string from the rock to his head and found it to be 8 feet. Does he have enough information to determine the height of the kite? If so, find the height of the kite. If not, state what other information would be needed. Yes, there is enough information. Let x represent the height DE. Then, = 150 x We are looking for the value of x that makes the fractions equivalent. Therefore, 8x = 825, and x = feet. The height of the kite is approximately 103 feet high in the air.