1. 8.4: Similarity in Right Triangles Objectives: Students will be able to… Find the geometric mean between 2 numbers Find and use relationships between similar right triangles.

2. Geometric Mean: In a proportion,, b and c are called the “means” Proportions in which the means are = occur frequently in geometry. THE GEOMETRIC MEAN, x:

3. Find the Geometric Mean of the following numbers: 1.2 and 10 2. 25 and 4

4. Definition: ALTITUDE of a triangle The altitude is a segment from a vertex of a triangle that is perpendicular to the opposite side (or the line containing the opposite side).

5. In a right triangle, the altitude to the hypotenuse creates 3 similar triangles. B D A A C B BD C

6. Right Triangle Similarity HYPOTENUSE LEG B LEG A ALTITUDE D C PROPORTIONS WHERE ALTITUDE AND LEGS ARE GEOMETRIC MEANS:

7. The length of the altitude of a right triangle is the geometric mean of the 2 pieces (segments) of the hypotenuse ALTITUDE D C Example: Find x. x 12 3

8. Examples: Find x. 1. 2. 3. 2 8 x x 3 9 50 40 x

9. The altitude to the hypotenuse of a right triangle separates the hypotenuse so that the leg is geometric mean of length of adjacent piece of hypotenuse and the whole length of hypotenuse. LEG B LEG A ALTITUDE D C Example: Find x. x 4 16

10. Examples. Find the value of x. 1. 2. 3. x 54 20 6 x x 36 60