Mean Proportional
in a Right Triangle
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"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.

 

Definition:
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.

 

Theorem:
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.

 

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

Altitude Rule:

 

 

Theorem:
Each leg of a right triangle is the mean proportional between the hypotenuse and the projection of the leg on the hypotenuse. 

or

Leg Rule:


 

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).

  
x2 = 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.

  
x2 = 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:

x2 = 4x + 16
x2 - 4x - 16 = 0
(Use Graphing Calculator to solve.  Graph and find where the graph crosses the x-axis.)
 


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