Notes Geometric Mean / Similarity in Right Triangles

Geometric Mean Geometric Mean is the square root of the product of two values. If a, b, and x are positive numbers and , then x is called the geometric mean between a and b. Example : Find the geometric mean of 3 and 12. = Write a proportion. 3 x 12 x2 = 36 Cross-Product Property Find the positive square root. x2 = 36 x = 6 The geometric mean of 3 and 12 is 6.

Theorem 9.1Similarity in Right Triangles If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. ∆CBD ~ ∆ABC, ∆ACD ~ ∆ABC, ∆CBD ~ ∆ACD

Similarity in Right Triangles - Corollary 1 seg1 alt seg2 The length of the altitude of the right triangle is the geometric mean between the segments of the hypotenuse .

Example Find the length of the altitude. 3 x = x 6 18 = x2 √18 = x

Similarity in Right Triangles – Corollary 2 leg SHAL hypotenuse Each leg of the right triangle is the geometric mean between the hypotenuse and the segment of the hypotenuse adjacent to the leg.

Example 3 Find the length of the leg. y 5 + 2 = y 2 7 y = y 2 14 = y2

Similarity in Right Triangles Solve for x. 2 6 6 x = Write a proportion. 2x = 36 Cross-Product Property x = 18