Definitions:

Right Triangle - A triangle that has a right angle in its interior.

Geometric Mean - The geometric mean of two positive numbers a and b is the positive number x such that a/x = x/b. If you solve this proportion for x, you will find that x equals the square root of (a) (b) which is a positive number.

Theorems

1. The altitude drawn to the hypotenuse of a right triangle creates two similar right triangles, each similar to the original right triangle and similar to each other.

2. If an altitude is drawn to the hypotenuse of a right triangle, then each leg is the geometric mean between the hypotenuse and its touching segment on the hypotenuse.

3. If an altitude is drawn to the hypotenuse of a right triangle, then it is the geometric mean between the segments of the hypotenuse.

Prove that two similar right triangles are similar.

Sample Problems:

1. Investigating Similar Triangles

A) Cut an index card along one of its diagonals.

B) On one of the right triangles, draw an altitude from the right angle to the hypotenuse. Cut along the altitude to form two right triangles.

C) You should now have three right triangles. Compare the triangles. What property do they share?

2. Geometric Mean

Example 1:

Example 2:

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