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Chapter 7: Similarity 7.4 Similarity in Right Triangles Review • Draw a right triangle. Label the triangle ABC, with C as the right angle. • Draw the altitude to the hypotenuse, and label it CD. • Name the two smaller right triangles that are formed. 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. A ABC ~ ACD ~ CBD D C B Geometric Mean • the number x such that: a x x b • where a, b, and x are positive numbers • example: • the geometric mean of 6 and 24 is 12: 6 x x 24 x 144 2 Example 1 • Find the geometric mean of 4 and 18. Example 1a • Find the geometric mean of 15 and 20. Corollary 1 to Theorem 7-3 • 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. A 6 D 24 12 C B Corollary 2 to Theorem 7-3 • 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 adjacent hypotenuse segment and the length of the hypotenuse. A 15 3 D 3 5 12 6 C 6 5 B Example 2 • Solve for x and y. x y 4 5 Example 3 • Solve for x and y. 4 x y 12 Example 4 • Solve for x: x 40 50 Example 5 • Find x, y, and z. z x y 1 4 Example 6 • Find x, y, and z. z x y 5 4 Example 7 • Find the value of x: x+2 5 x Example 8 • Find the value of x: x+3 x 12 Example 9 • Find the value of x: 2x + 1 x 8 Example 10 • Find the values of the variables: 6 30 z y x Homework • p. 394: • 2-20 even, 26-36 even