Similarity in Right Triangles Lesson 8.1

Pre-AP Geometry

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.

Simplifying Radicals

Product Rule for Radicals

For any nonnegative numbers a and b and any natural number index n,

n a



n b



n ab

Quotient Rule for Radicals

For any natural number index n and any real numbers a and b (b 

n a b



n a n b

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 n th power of an integer or polynomial.

The radicand contains no fractions.

No radicals appear in the denominator.

Simplifying Radicals Example 1: 75 Example 2: 2 5 Example 3: 2 48 Example 4: 16  4

Geometric Mean

a x

If a, b, and x are positive numbers with , then x is the geometric mean between a and b.

x



b

Geometric Mean Example 5: Find the geometric mean of 4 and 9.

Example 6: Find the geometric mean of 3 and 48.

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.

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 C A D B

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.

C

AD CD



CD DB

A D B

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.

C

AB



AC AC AD AB BC



BC BD

A D B

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

A D B

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

A D B

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

Written Exercises Problem Set 8.1B, Handout 8-1