Similarity in Right Triangles

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Presentation transcript:

Similarity in Right Triangles

Right Triangle Relationships Theorem 7-3: 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.

Identifying Similar Triangles What similarity statement can you write relating the three triangles in the diagram?

 What similarity statement can you write relating the three triangles in the diagram if the “big” triangle is QRP?

Geometric Mean For any two positive integers a and b, the geometric mean of a and b is the positive number x such that .

Finding geometric means Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 4 and 25

Finding geometric means Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 5 and 30

Geometric Means in Right Triangles Corollary 1: The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse.

Geometric Means in Right Triangles Corollary 2: The altitude to the hypotenuse of a right triangle separates the hypotenuse so that the length of each leg of the triangle is the geometric mean of the length of the hypotenuse and the length of the segment of the hypotenuse adjacent to the leg.

Using the Corollaries What are the values of x and y?

What are the values of x and y?

Finding a Distance You are preparing for a robotics competition using the setup shown. You program the robot to move from A to D and pick up the plastic bottle. How far does the robot travel from A to D?