# Mean Proportional

Mean proportional is another term for geometric mean. Geometric mean is a measure of central tendency, like the arithmetic mean. It is calculated differently, however. It is a common measurement in ratios and proportions.

Index

## How to Calculate Mean Proportional

Here, we are going to discuss how the mean proportional is defined, and how to calculate it.

### Definition and Formula

For two real, positive numbers $$a$$ and $$b$$, the geometric mean (also known as mean proportional) is the number $$x$$ satisfying:

$$\frac{a}{x} = \frac{x}{b}$$

In other words,

$$x = \sqrt{ab}$$

Note that the term inside the square root must be positive for this formula to work.

### Mean Proportional Between Two Numbers

As we have seen above, to calculate the mean proportional between two numbers $$a$$ and $$b$$ we have to:

1. Take the product of the two numbers, $$ab$$
2. Take the square root of this product.

This gives us the mean proportional between the two numbers, or the geometric mean.

## Applications of Mean Proportional

### Geometric Mean Altitude Theorem

In a right angled triangle ABD, if we drop a perpendicular AC onto hypotenuse BD, we have:

$$\angle ACD = \angle DAB = 90 \circ$$

As well as,

$$\angle ADB = \angle BAC$$

Thus, by AA similarity criterion,

$$\triangle ADB \sim \triangle CDA \sim \triangle CAB$$

This gives us, by definition of similarity,

$$\frac{CD}{AC} = \frac{AC}{CB}$$

In other words, AC is the geometric mean of CD and CB.

$$AC = \sqrt{CD\cdot CB}$$

We can get CD and CB from similar relations:

$$\frac{CD}{DA} = \frac{DA}{DB}$$

Or,

$$CD = \frac{{(DA)}^2}{DB}$$

Similarly,

$$CB = \frac{{(AB)}^2}{DB}$$

## Solved Examples

Question 1. Calculate the mean proportional between 234 and 104.

Solution. Here, we can write $$a$$ = 234, $$b$$ = 104. Then, by definition of geometric mean

$$x = \sqrt{ab}$$

Substituting $$a$$ and $$b$$, we have,

$$x = \sqrt{234 \cdot 104}$$

$$x = \sqrt{24336} = 156$$

Thus, the geometric mean of 234 and 104 is 156.

Question 2. Calculate the length of altitude on hypotenuse, for a triangle with sides 3 cm, 4 cm and 5 cm.

Solution. Using above figure again for reference, let AB = 3 cm, AD = 4 cm, DB = 5 cm.

From Geometric Mean Altitude Theorem, we know that the altitude AC is given by,

$$AC = \sqrt{CD \cdot CB}$$

We obtain CD and CB separately as follows:

$$CD = \frac{{(DA)}^2}{DB}$$

$$CD = \frac{16}{5} = 3.2$$

And,

$$CB = \frac{{(AB)}^2}{DB}$$

$$CB = \frac{9}{5} = 1.8$$

So we now have altitude AC as,

$$AC = \sqrt {1.8 \cdot 3.2} = \sqrt{5.76} = 2.4$$

Thus, the length of the altitude is 2.4 cm.

## FAQs

How do you find the mean proportional of A and B?

The mean proportional x of A and B is defined as (x = \sqrt{A \cdot B}).

What is the mean proportional of 4 and 9?

The mean proportional or geometric mean $$x$$ of 4 and 9 is given by,
$$x = \sqrt{4 \cdot 9} = \sqrt{36}$$
$$x = 6$$
Thus, 6 is the geometric mean of 4 and 9.

What is the formula for continued proportion?

The ratios a:b and b:c are in continued proportion if,
$$\frac{a}{b} = \frac{b}{c}$$
Then c is called the third proportion of a and b, while b is the geometric mean of a and c.

Find the third proportional of 16 and 32.

Let x be the third proportional of 16 and 32. In other words, 32 is the mean proportional of 16 and x. Then we have,
$$\frac{16}{32} = \frac{32}{x}$$
Or,
$$x = \frac{32^2}{16} = 64$$
Thus, the third proportional of 16 and 32 is 64.

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