"Mean Proportional"
may also be referred to as
a "Geometric Mean". 
Remember the rule for working with proportions:
the product of the means equals the product of
the extremes.



In a mean proportional problem, the "means"
are the same
values. 


The mean proportional of two positive numbers
a and
b is the
positive number
x such that
.
When solving,
. 
Notice that the x value appears TWICE in the "means"
positions. 

The altitude to
the hypotenuse of a right triangle forms two triangles that
are similar to each other and to the original triangle.


Since these
triangles are similar, we can establish
proportions relating the corresponding sides.
Two valuable theorems can be formed using these
proportions. 


The
altitude to the hypotenuse
of a right triangle is the mean proportional between the
segments into which it divides the hypotenuse. 

Each
leg of a right triangle
is the mean proportional between the hypotenuse and the
projection of the leg on the hypotenuse. 
Examples:
1. 
Find
x:

Solution: Examine the diagram to see
what is given.
This problem needs the Altitude Rule. 
4x = 64
x = 16 
2. 
Find
x:

Solution: Examine the diagram to see
what is given.
This problem needs the Leg Rule (there is no value on the
altitude). 
x^{2} = 16
x = 4
(lengths are positive) 
3. 
Find
x to
nearest tenth:

Solution: Examine the diagram to see
what is given.
This problem needs the Leg Rule (there is no value on the
altitude). But be careful in this problem. You will
need the ENTIRE hypotenuse length which is 4 + 12 = 16. 
x^{2} = 192
x = 13.9
(lengths are positive) 
4. 
In a starfish, the
outer leg length, AB, is the mean proportional between the
length of the base of the
leg, BC, and the length of the outer leg plus the base,
AC. The base
of the leg is 4 cm. Find the length of the outer leg.
Use your graphing calculator to solve the quadratic equation,
rounding your answer to the nearest tenth.

Solution:
x^{2 }= 4x + 16
x^{2 } 4x  16 = 0
(Use Graphing Calculator to solve.
Graph and find where the graph crosses the xaxis.)

Use 2nd Calc #2 Zero to find the positive root.
x = 6.472136
Outer leg = 6.5 cm 

