Similarity in Right Triangles

Math 2 - Lesson 32Mr. Lopez

You Can Do It!!

Given: AB and BD are

Altitudes.

Prove: BDC ~ ABC

A

B C

D

Facts about altitudes and right triangles

• Did you know? 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!

(SUPER COOL! )

A

B CD

ABC ~ ADC ~ ABD

WOW!!!

Geometric mean. (Who’s a meanie?)

• The geometric mean is the result when given the two positive numbers (“a” and “b”) such that:

or

b

x

x

a

abx

Huh? Let’s try it!

• Find the geometric mean of 4 and 18.

• Find the geometric mean of 15 and 20.

Theorem using geometric mean• The length of the altitude to the hypotenuse of a right

triangle is proportional with the lengths of the segments that make up the hypotenuse. (WHAT?!?)

CDBDADAD

A

B CD

AD

CD

BD

AD

Another theorem using geometric mean

• The altitude to the hypotenuse of a right triangle separates the hypotenuse so that the length of each leg of the triangle is proportional with the length of the adjacent hypotenuse segment and the length of the hypotenuse. (OMG!)

BCADACAC

C

A BD

AB

AC

AC

AD

So Stressed! Let’s Practice

• Solve for “x” and “y”

A

B CD

yx

45

Kinda Understand! Try Again!

• Solve for x, y, and z.

A

B CD

y8

x12

z

Homework Answers