5.3 Properties of Right Triangles

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Presentation transcript:

5.3 Properties of Right Triangles CHAPTER 5.3 Properties of Right Triangles Copyright © 2014 Pearson Education, Inc.

Theorem 5.13 Altitude of a Right Triangle Theorem The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and to each other. Copyright © 2014 Pearson Education, Inc.

Writing Similarity Statements What similarity statement can you write relating the three triangles in the diagram? Solution To help, sketch the triangles separately in the same orientation. Segment YW is the altitude to the hypotenuse of right ΔXYZ. Using Theorem 7.5-1. There are three similar triangles. ΔXYZ ∼ ΔYWZ ∼ ΔXWY Copyright © 2014 Pearson Education, Inc.

Corollary 5.14 Geometric Mean in Similar Right Triangles: Hypotenuse Copyright © 2014 Pearson Education, Inc.

Corollary 5.15 Geometric Mean in Similar Right Triangles: Legs Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Theorem 5.16 The median from the right angle in a triangle is one-half the length of the hypotenuse. Copyright © 2014 Pearson Education, Inc.

Using the Corollaries and Algebra What are the values of x and y? Solution To find the altitude y, we can use Corollary 1. After that, since x is a leg of the right triangle, let’s try Corollary 2. Copyright © 2014 Pearson Education, Inc.

Using the Corollaries and Algebra What are the values of x and y? To find x. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Finding a Distance You are preparing for a robotics competition using the setup shown here. Points A, B, and C are located so that AB = 20 in. and Point D is located on segment AC so that and DC = 9 in. You program the robot to move from A to D and to pick up the plastic bottle at D. How far does the robot travel from A to D? Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Finding a Distance Solution Concentrate on right ΔABC with altitude .We try Corollary 2 since we are not trying to find the altitude length BD. Only the positive solution makes sense in this situation. The robot travels 16 in. Copyright © 2014 Pearson Education, Inc.