8-48-4Similarity in Right TrianglesSimilarity in Right Triangles

One Key TermOne Key TermOne TheoremOne Theorem

Two CorollariesTwo Corollaries

Draw a diagonal across your index card.Draw a diagonal across your index card.

On one side of the card use a ruler to draw On one side of the card use a ruler to draw the altitude of the right triangle from the the altitude of the right triangle from the corner of the index card perpendicular to corner of the index card perpendicular to the diagonal.the diagonal.

Cut out the Cut out the

three triangles, three triangles,

examine themexamine them

Theorem 8-3 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.

abx

b

x

x

a

2

abx

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

300

20

15

2

x

x

x

300x

310x

1. The geometric mean can give a meaningful "average" to compare two companies.

2. The use of a geometric mean "normalizes" the ranges being averaged, so that no range dominates the weighting.

3. The geometric mean applies only to positive numbers.[2]

4. It is also often used for a set of numbers whose values are meant to be multiplied together or are exponential in nature, such as data on the growth of the human population or interest rates of a financial investment.

Purpose of the Geometric Mean

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.

DB

CD

CD

AD

A

C

BD

)(DBADCD

Corollary 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.

,AB

AC

AC

AD

A

C

BD

AB

CB

CB

BD

Similarity in Right TrianglesFind the values of x and y in the following right triangle.

4 5

X Y Y

X

4 + 5

9

4 x

x

You Try One!!!Find the values of x and y in the following right triangle.

#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