Mean proportional appearing in the extreme positions
The mean proportional between two numbers or quantities B and C is a value X that satisfies the proportion $$ X:B = C:X $$ meaning that X appears in both extreme positions of the proportion.
In a proportion, the first and last terms are called the extremes, while the middle terms are called the means.
$$ \color{red}X:B = C:\color{red}X $$
This is a special type of proportion in which the same value occupies both extreme positions.
A practical example
In the following proportion, the mean proportional is 9.
$$ 9:27 = 3:9 $$
This is because the number nine appears in both extreme positions.
$$ \color{red}9:27 = 3:\color{red}9 $$
How to compute the mean proportional
In a proportion of the form $$ X:B = C:X $$ the mean proportional is equal to the square root of the product of the mean terms, that is $$ X = \sqrt{BC} $$
This formula provides a direct and efficient way to compute the mean proportional.
Alternatively, the same result can be obtained by following the full sequence of algebraic steps shown in the proof, although this approach is more time-consuming.
The proof
Consider a proportion in which the same value appears in both extreme positions (X).
$$ X:B = C:X $$
Apply the fundamental property of proportions and compute the cross product of the terms.
$$ X \cdot X = B \cdot C $$
$$ X^2 = BC $$
Apply the invariant property of equations to take the square root of both sides.
$$ \sqrt{X^2} = \sqrt{BC} $$
Then simplify √X2=X using radical reduction.
$$ \sqrt[\not{2}]{X^{\not{2}}} = \sqrt{BC} $$
It follows that the mean proportional is equal to the square root of the product of the mean terms.
$$ X = \sqrt{BC} $$
This establishes the formula for the mean proportional.
Example 1. Consider the proportion $$ x:2 = 32:x $$ The mean proportional x is the square root of the product of the mean terms $$ x = \sqrt{2 \cdot 32} = \sqrt{64} = \pm 8 $$ Therefore, the mean proportional is 8. $$ 8:2 = 32:8 $$ To verify this, compute the cross product $$ 8 \cdot 8 = 32 \cdot 2 $$ $$ 64 = 64 $$ Since both sides are equal, the proportion is verified.
Example 2. Alternatively, compute the mean proportional by carrying out all the algebraic steps. $$ x:2 = 32:x $$ Apply the fundamental property of proportions and compute the cross product of the terms $$ x \cdot x = 2 \cdot 32 $$ $$ x^2 = 64 $$ Apply the invariant property of equations to take the square root of both sides $$ \sqrt{x^2} = \sqrt{64} $$ $$ x = \pm 8 $$ The mean proportional is 8. $$ 8:2 = 32:8 $$ This is the same result obtained using the direct formula.
And so on.
