7.1 Geometric Mean

1. 7.1 Geometric Mean What you’ll learn: To find the geometric mean between 2 numbers To solve problems involving relationships between parts of a right triangle and the altitude to its hypotenuse.

2. Geometric Mean For 2 positive numbers a and b, the geometric mean is the positive number x where the proportion a:x=x:b is true, also written as or with cross products as The geometric mean between 2 numbers is the positive square of their product. Ex: find the geometric mean between each pair of numbers. • 2 and 50 • 25 and 7

3. Theorem 7.1 If the altitude is drawn from the vertex of the right angle of a right triangle to its hypotenuse, then the 2 triangles formed are similar to the given triangle and to each other. ADB~BDC ADB~ABC CDB~CBA A D C B

4. Theorem 7.2 The measure of an altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse is the geometric mean between the measures of the 2 segments of the hypotenuse. a f e d b c

5. Theorem 7.3 If the altitude is drawn from the vertex of the right angle of a right triangle to its hypotenuse, then the measure of a leg of the triangle is the geometric mean between the measures of the hypotenuse and the segment of the hypotenuse adjacent to that leg. a f e d b c

6. Find x, y, and/or z 2 x y y x 8 14 1. 2. 3. 4. 4 A D A A D D A D z 10 17 y x y 20 C C B x 6 B C C B z B

7. Homeworkp.34614-34 evenQuiz tomorrow on 7.1