Similarity in Right Triangles 7-4

Warmup activity (don’t need to turn in) Complete activity on p. 391 with a partner

Theorem In a right triangle, the altitude to the hypotenuse yields three similar triangles. A B C D

Example a d b c e f a f b d a b c e e

Write an extended proportion for the sides in the medium and small triangles. A C B D

Geometric Mean Proportions in which the means are equal occur frequently in geometry. For any two positive numbers a and b, the geometric mean of a and b is the positive number such that Note that

Examples – Finding the geometric mean Find the geometric mean of the following numbers (round to two decimal places): a. 4 and 18b. 15 and 20c. 3 and 12

Corollary 1 The lengths of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse.

Proof of Corollary 1 *Bonus* Prove

Examples Find the missing variables: x x4 6

Corollary 2 When the altitude to the hypotenuse of a right triangle separates the hypotenuse, the length of each leg is the geometric mean of the length of the adjacent hypotenuse segment and the length of the hypotenuse.

Applying Corollaries 1 and 2 Solve for x and y and z.You try x y 45 z x 12 z 4 y

Hwk P : 1-20, 34-36,49-51