“What we do in life echoes in eternity.”

7.4 Similarity in Right Triangles

LT: To find and use relationships of similar right triangles.

Similarity in Right Triangles

Theorem 7-3: The altitude to the hypotenuse of a right triangle divides the triangles into two triangles that are similar to the original triangle and to each other.

Geometric Mean

Geometric Mean: The number x such that , where a, b, and x are positive numbers

a

xx

b

Find the geometric mean of 3 and 27.

Review: How do we find the arithmetic mean of 3 and 27?

Note:

x ab

Find the geometric mean of 4 and 18.

The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers.

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

Geometric Mean

5.2 in 8.75in

6.75in

Corollary to Theorem 7-3: The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse

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

4 5

X Y Y

X

4 + 5

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

“You wasted $150,000 on an education you coulda got for $1.50 in late fees at the public library.”

7.4 Similarity in Right Triangles

HW (7.4) Pgs. 394-396: #1-21, 34, 35, 50, 51

Proof of Corollary to Theorem 7-3

A

C

D B

Statements Reasons

1.

2.

3.

1.

2.

3.

Right triangle, ABC, with

CD the altitude to the hypotenuse

AD

CDCD

DB

Given : Right triangle, ABC, with

CD the altitude to the hypotenuse

Prove : AD

CDCDDB

Altitude of rt. Δ to hypotenuse divides into 2 ~ Δs

Real World ConnectionAs Marla arrives at the lake from the parking lot, she reads a sign that says she is 320m from the dock. How far is Marla from the information center?

Kick it up a notch!Find the value of x in the following right triangle.

x1

2x - 1

Similarity in Right Triangles

m1m4 m7

m2 m6 m8

m3 m5 m9