Name: _____________________
Ms. Anderle
Date: __________ Class: ________
Geometric Mean
Geometric Mean
Geometric Mean: When the means of a proportion are the same number, that number is
called the _____________________________ of the extremes. The geometric mean between
two numbers is the positive square root of their product.
extreme  a = x  mean
mean  x
= b  extreme
Basically, the geometric mean of two positive numbers a and b is the number x
a x
such that
 . So, x2 = ab and x  ab .
x b
Example: The geometric mean of a = 9 and b = 4 is ___________.
Why?
Now try on your own:
Find the geometric mean between each pair of numbers.
1) 5 and 45
2) 12 and 15
3) 2 and 8
Homework:
For each of the following, find the geometric mean. SHOW ALL WORK!!!!
1) 245 and 500
2) 120 and 10
3) 200 and 40
4) 50 and 25
5) 12.5 and 30
6) 11,000 and 25
7) 65 and 100
8) 20.5 and 50
9) 44 and 61
10) 76 and 54
11) 762 and 65
12) 5 and 14
Name: _____________________
Ms. Anderle
Date: __________ Class: ________
Geometric Mean in Triangles
Geometric Mean in Right Triangles
The geometric mean is present in right triangles. In these triangles we call it the mean
proportional.
In a right triangle, an altitude drawn from the vertex of the right angle to the
hypotenuse forms two additional right triangles. These three right triangles share a
special relationship!
Theorem: The altitude to the hypotenuse of a right triangle is the mean proportional
between the segments into which it divides the hypotenuse.
If WZ is altitude to hypotenuse UV of right triangle UVW, then….
We call this the altitude rule….
part of hypotenuse
altitude
=
altitude
other part of hypotenuse
Theorem: Each leg of a right triangle is the mean proportional between the hypotenuse
and the projection of the leg on the hypotenuse.
leg
leg
projection
projection
If CD is altitude to hypotenuse AB of right triangle ABC, then….
We call this the leg rule:
hypotenuse = leg
leg
projection
Example:
1) Solve for x:
Which rule do we need to use? __________________________
2) Solve for x:
Which rule do we need to use? __________________________
3) Find x (to the nearest tenth)
Which rule do we need to use? __________________________
4) In right triangle ABC, angle C is a right angle and CD is the altitude to AB. If AD = 6
and BD = 10, find CD.
Homework
1) In triangle ABC, angle C is a right angle and CD is the altitude to AB. If AC = 6
and AB = 9, find AD.
2) In right triangle ABC, the right angle is at C and altitude CD is drawn to
hypotenuse AB. If AD = 3 and DB = 9, find AC.
3) In a right triangle whose hypotenuse measures 50, the shorter leg measures 30.
Find the measure of the projection of the shorter leg on the hypotenuse.
More Practice with Mean Proportional:
Use the following diagram for # 1 – 4.
1) If AC = 6 and AB = 9, find AD
2) If AC = 5 and AB = 9, find AD
3) If AD = 2 and AB = 18, find AC.
4) If AD = 3 and DB = 9, find AC.
5) Find CD (BE CAREFUL!!!)