Similar Triangles, Pythagorean Theorem, and Congruent Triangles.

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1 ay 20 Teacher Page Similar Triangles, Pythagorean Theorem, and ongruent Triangles. Pythagorean Theorem Example 1: circle has a radius of 20 units. triangle is formed by connecting a point on the perimeter of the circle with the endpoints of the diameter. If the shortest side of the triangle is twelve units long, how long is the third side? Explain how you found your answer. 12 Similar Triangles Example 2: man 6 ft tall casts a shadow that is 11 ft. 6 in. long. The end of his shadow coincides with the end of the shadow cast by a building 128 ft. from the man. Find the height of the building. Explain how you found your answer. 6 ft 128 ft 11 ft 6 in ongruence in Triangles Example 3: To find the distance across this lake, a surveyor located points and with, P and collinear. P= P and P=. Explain why distance is equal to distance. P

2 Name: ate: Per: ay 20 Similar Triangles, Pythagorean Theorem, and ongruent Triangles. 1. n engineer designing a bridge to be built across a deep canyon needs to find the distance across the canyon. He locates points P and so that P=P. Next he uses a transit to measure < P. He then locates point so that m<p=m<p. How can the engineer find the distance? Explain how you found your answer. P 2. When a mirror is placed on the ground so that the top of a building can be seen beside a person standing by the mirror, m<m=m<m. person 150 cm tall who is 6 m from the mirror observes the top of the tower when the mirror is 120 m from the tower. Find the height of the building. Explain how you found your answer. M 3. vacant rectangular city lot is twice as long as it is wide. What is the diagonal length of the lot if the longest side is 528 feet? Explain how you found your answer.

3 4. To find the distance x across a river, a surveyor located points,,, and through direct measurement. Find the distance x. tree x 25 m 75 m 100 m 5. To stabilize a tall pole, a guy wire is stretched from the top of the pole to a point on the ground that is one-third the height of the pole away from the base of the pole. How long must the guy wire be to stabilize a 24 ft pole? Guy wire 6. cat is stuck 35 feet up a tree. firefighter sits his ladder 8 feet away from the tree to reach it.. How long is the ladder?. If the cat climbs up another 5 feet, will the firefighter still be able to reach the cat if he moves his ladder closer to the tree. 7. The SI team wants to determine the height of a bad guy in a photo next to a stop sign. They know the actual stop sign is 8 feet tall. In the photo, the stop sign is 5 inches tall and the bad guy is 4 inches tall. Use this information to help the SI team to determine the actual height of the bad guy. Show and explain your work.

4 8. Roger wants to find the height of the flagpole. Roger is 5 feet 6 inches tall and casts a shadow about 3 feet long. The flagpole casts a shadow about 11 feet long. Use this information to help Roger find the approximate height of the flagpole. 1) What proportion can be used to find the height of the flagpole? 2) Find the height of the flagpole. 9. Mary needs to clean the windows on the second floor of her house. The ladder is placed 4 feet from the house. The windows are 15 feet up. How long does the ladder need to be to reach the window? Round your answer to the nearest tenth if necessary foot pole is upright. stake is placed 9 feet from the base of the pole. How much rope is needed to brace the pole? Round your answer to the nearest tenth if necessary. Fred's House 11. How far would Fred have to drive if he went to ill s house by way of licia s house? 16 miles licia's House 12. ngelica and James leave their school at the same time. ngelica drives due east for five miles. James drives due north. They drive until they are 17 miles apart. How far did James drive? 3.5 miles ill's House

5 ay 20 Key Example x = 40 The third side is 38.2 units long. Example 2 x 1674 = The tree is 72.8 ft tall. Example 3 The m P = m P by the Vertical ngles Theorem. The triangles are congruent by the SS (Side ngle Side) ongruence Postulate. Since the triangles are congruent that makes = because they are parts of congruent triangles by PT (orresponding parts of congruent triangles are congruent). Student Handout 1. The m P = m P by the Vertical ngles Theorem. The triangles are congruent by the S (ngle Side ngle) ongruence Postulate. Since the triangles are congruent that makes = because they are parts of congruent triangles by PT (orresponding parts of congruent triangles are congruent). x = The tree is 30 m tall = x = x The length of the diagonal is ft. x = x The distance across the river is 75 units = x The guy wire must be 25.3 ft long. 6..) = x The ladder is 35.9 ft long..) The cat would be 40 ft high and the ladder is only 35.9 ft so the firefighter would not be able to reach the cat x = The bad guy is 6.4 feet tall.

6 x = 11 The flagpole is 20.2 feet tall = x The ladder needs to be 15.5 feet to reach the window = x The rope needs to be 15 feet long to brace the pole. 11. x = 16 From licia s house to Fred s house is 15.6 miles. Fred has to drive 19.1 miles to get to ill s house. 12. x + 5 = 17 James drove 16.2 miles.

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