Similarity in Right Triangles

Lesson 8.1Pre-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,

Quotient Rule for Radicals

For any natural number index n and any real numbers a and b (b 0) where and are real numbers,

n n na b ab

n a n b

n

nn

a a

b 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 nth

power of an integer or polynomial.• The radicand contains no fractions.• No radicals appear in the denominator.

Simplifying Radicals

Example 1:

Example 2:

Example 3:

Example 4:

75

2

5

2 48

16 4

Geometric Mean

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

This implies that .

a x

x b

x ab

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

A B

C

D

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.

A B

C

D

AD CD

CD DB

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.

A B

C

D

AB AC

AC AD

AB BC

BC BD

Similarity in Right Triangles

Example 7: If BD = 16 and AD = 9, find CD, AB, CB, and AC.

A B

C

D

Similarity in Right Triangles

Example 8: If BD = 4 and CD = 2, find AD, AB, CB, and AC.

A B

C

D

Written Exercises

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

Written Exercises

Problem Set 8.1B, Handout 8-1