Chapter 8 Lesson 4

Objective:Objective: To find To find and use and use

relationships in relationships in similar right similar right

trianglestriangles..

Theorem 8-3  The altitude to the hypotenuse of a

right triangle divides the triangle into two triangles that are similar to the original triangle and to each other.

AltitudeAltitude

The geometric mean is the number x such that   =  , where a, b and x are positive numbers.

xa

bx

Example 1: Finding Geometric Mean

Find the geometric mean of 4 and 18.     

The geometric mean of 4 and 18 is 6 .

2

What they What they ask for ask for 11stst..

What they What they ask for ask for 22ndnd..

184 xx

722 x72x

236x26x

Example 2: Finding Geometric Mean

Find the geometric mean of 15 and Find the geometric mean of 15 and 20. 20.

2015 xx

3002 x300x

3100x310x

Corollary 1 to Theorem 8-3The length of the altitude to the hypotenuse of a right triangle is the geometric mean of

the lengths of the segments of the hypotenuse.

AltitudeAltitude

WTSW

SWRW

Corollary 2 to Theorem 8-3Corollary 2 to Theorem 8-3The altitude to the hypotenuse of a right triangle separates the hypotenuse so that the length of

each leg of the triangle is the geometric mean of the length of the adjacent hypotenuse segment

and the length of the hypotenuse.

∆ACD ~ ∆ABC

ABAC

ACAD

∆CBD ~ ∆ABC

ABCB

CBDB

Example 3: Applying Corollaries 1 and 2

Use Corollary 2 to solve for x:

544

xx

362 x36x

6x

Use Corollary 1 to solve for y:

45 yy

202 y20y

54 y52y

Example 4: Applying Corollaries 1 and 2

Solve for x and y.               

Use Corollary 2 to solve for x:

1244

xx

642 x64x8x

Use Corollary 1 to solve for y:

412 yy

482 y48y

316y34y

AssignmenAssignmentt

Pg. 442 Pg. 442 #1-20; 26-36#1-20; 26-36