1. Similarity in Right Triangles
Lesson 8.1
Pre-AP Geometry

2. Lesson Focus
Right triangles have many interesting properties. This lesson begins with a study of the properties of right triangles. Algebraic skills with radicals are reviewed and are used throughout the chapter.

3. Simplifying Radicals
Product Rule for Radicals For any nonnegative numbers a and b and any natural number index n, Quotient Rule for Radicals For any natural number index n and any real numbers a and b (b  0) where and are real numbers,

4. Simplifying Radicals
Radical expressions are written in simplest terms when:
The index is as small as possible
The radicand contains no factor (other than 1) which is the nth power of an integer or polynomial.
The radicand contains no fractions.
No radicals appear in the denominator.

5. Simplifying Radicals
Example 1: Example 2: Example 3: Example 4:

6. Geometric Mean
If a, b, and x are positive numbers with , then x is the geometric mean between a and b. This implies that .

7. Geometric Mean
Example 5: Find the geometric mean of 4 and 9. Example 6: Find the geometric mean of 3 and 48.

8. Important
To be successful in this chapter, you should spend the time to memorize the following theorem and corollaries. It is very important to your success in this chapter to remember these rules and know how use them.

9. Right Triangle Similarity Theorem
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.
ACB  ADC  CDB

10. Corollary 1
When the altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the geometric mean between the segments of the hypotenuse.

11. Corollary 2
When the altitude is drawn to the hypotenuse of a right triangle, each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse that is adjacent to that leg.

12. Similarity in Right Triangles
Example 7: If BD = 16 and AD = 9,
find CD, AB, CB, and AC.

13. Similarity in Right Triangles
Example 8: If BD = 4 and CD = 2,
find AD, AB, CB, and AC.

14. Problem Set 8.1A, p.288: # 2 - 38 (even)
Written Exercises
Problem Set 8.1A, p.288: # (even)

15. Written Exercises
Problem Set 8.1B, Handout 8-1