**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:

- Take the product of the two numbers, \(ab\)
- 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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