Section 8-4 Similarity in Right Triangles SPI 31A: identify corresponding parts of similar and congruent geometric figures

Objectives:• Find and use relationships in right triangles

Geometric Mean

If you drop a perpendicular from the right angle of a right triangle to the opposite side, you will three similar right triangles. This altitude is known as the geometric

mean.

Right Angle

Geometric mean(altitude)

Theorem 8-3The 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.

Corollary 1 to TheoremThe length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segment of the hypotenuse.

a b

x y or y = ab

a y

y b

Geometric Mean Formula

Finding the Geometric Mean

For any two positive numbers a and b, the geometric mean of a and b is the positive number y such that,

and y = aba y

y b

Find the geometric mean of 4 and 18.

4

18

x

x

a b

y

Corollary 2 to TheoremThe altitude of 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.

a b

x y

a x

x a b

Real-World ConnectionThe 300m path to the information center and the 400m path to the canoe rental dock meet at a right angle at the parking lot. Marla walks straight from the parking lot to the lake as shown. How far is Marla from the information center?

Hint: Use Corollary 2

AD AC

AC AB

300

300 500

AD

Solve for AD

AD = 180Marla is 180m from the info center

Explore the Geometric Mean