8.1: Geometric Mean

• Determine the geometric mean between two numbers.

• Solve problems using the geometric mean involving relationships between parts of a right triangle

Objectives: I will be able to….

Remember this??

a

b

c

d

Extremes

Means

8.1

Geometric Mean

Find the geometric mean between 2 and 50.

Answer: The geometric mean is 10.

Definition of geometric mean

Let x represent the geometric mean.

Cross products

Take the positive square root of each side.

Simplify.

A. 3.9

B. 6

C. 7.5

D. 4.5

A. Find the geometric mean between 3 and 12.

On your worksheet, you found the 3 types of geometric means that can occur when you have a right triangle with an altitude drawn from the right angle.

c c

c

Use Geometric Mean with Right Triangles

Find c, d, and e.

Since e is the measure of the altitude drawn to the hypotenuse of right ΔJKL, e is the geometric mean of the lengths of the two segments that make up the hypotenuse, JM and ML.

Use Geometric Mean with Right Triangles

Find c, d, and e.

Since d is the measure of leg JK, d is the geometric mean of JM, the measure of the segment adjacent to this leg, and the measure of the hypotenuse JL.

Use Geometric Mean with Right Triangles

Find c, d, and e.

Since c is the measure of leg KL, c is the geometric mean of ML, the measure of the segment adjacent to KL, and the measure of the hypotenuse JL.

• So, what is geometric mean useful for?

• There are many real life situations that we can use it to help us solve problems involving right triangles!

• VIEWING ANGLE A photographer wants to take a picture of a beach front. His camera has a viewing angle of 90° and he wants to make sure two palm trees located at points A and B in the figure are just inside the edges of the photograph.

 He walks out on a walkway that goes over the ocean to get the shot. If his camera has a viewing angle of 90°, at what distance down the walkway should he stop to take his photograph?

Answer: 60 ft.

A. 68.3 ft

B. 231.3 ft

C. 273.1 ft

D. 436.1 ft

AIRPLANES A jetliner has a wingspan, BD, of 211 feet. The segment drawn from the front of the plane to the tail, at point E. If AE is 163 feet, what is the length of the aircraft to the nearest tenth of a foot?