8-18-1Similarity in Right TrianglesSimilarity in Right Triangles

One Key TermOne Key TermOne TheoremOne Theorem

Two CorollariesTwo Corollaries

Theorem 8-1 Altitude Similarity Theorem

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.

CBDACDABC ~~

A

C

BD

Vocabulary1. Geometric Mean 1.

b

x

x

a

abx

#1 Finding the Geometric Mean Find the geometric mean of 15 and 20.

20

15 x

x

)20(15x

300x

310x

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

DB

CD

CD

AD

A

C

BD

)(DBADCD

Corollary 2 to Theorem 8-1The 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.

,AB

AC

AC

AD

A

C

BD

AB

CB

CB

BD

#2

x

4

12y

16

4 x

x

12

4 y

y

642 x 482 y

8x 34y

• Solve for x and y.

Small

Medium

Large

Leg Small Leg Large Hypotenuse

4 x

12y

x 16

y