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The term mean proportion is also referred to as the Geometric mean. The term means when used alone or in context with mean, median, or mode refer to the Arithmetic mean or finding an average. Geometric Mean or Mean Proportional is not similar to Arithmetic mean. In Mathematics, Arithmetic means deals with addition, whereas Geometric means deals with multiplication. Let us understand what the mean proportion is in terms of ratio and proportion.

In Mathematics, the mean proportion between two terms of a ratio is calculated by taking the square root of the product of two quantities in a ratio. For example, in the proportion p:q :: r:s, we can calculate the mean proportion for the ratio p:q by calculating the square root of the product of the quantity p and q. Mathematically, the mean proportion is expressed as:

Mean Proportion - \[\sqrt{pq}\]

The mean proportion or geometric mean of two positive numbers p and q is the positive number x , such that p/x = x/q. When solving the variable, x = \[\sqrt{p.q}\].

Note: The geometric mean or mean proportion together with the values is always positive.

In “mean proportion”, or “geometric mean” both means x in p/x = x/q, have the same values.

Find the mean proportional between 4 and 25.

Solution:

Let the mean proportion between 4 and 25 is x.

Accordingly,

\[x^{2} = \sqrt{4 \times 25}\]

\[x^{2} = \sqrt{100}\]

x = 10

Therefore, the mean proportion between 4 and 25 is 10.

The geometric mean or mean proportional with a right-angled triangle appears with two popular theorems. Let us understand the mean proportional theorem in terms of right-angled triangles.

Theorem 1: The altitude that is drawn to the hypotenuse of a right-angled triangle creates two triangles that are similar to the original triangle and each other.

As per the theorem, In right ABC, with altitude CD, the following relations can be established.

△ADC 〜 △CDB

△ACB 〜 △ADC, and

△ACB 〜 △CDB

As the triangles are similar, we can establish a proportional relationship between them. Two valuable theorems can be found using 3 of the proportions given below:

AB/AC = AC/AD, AB/CB = CB/DB, AD/CD = CD/DB

The altitude to the hypotenuse of a right-angled triangle is the mean proportional between the left and right parts of the hypotenuse of the right-angled triangle.

Mathematically, the altitude rule says,

[One Part of Hypotenuse (Left)]/Altitude= Altitude/[Other Part of Hypotenuse (Right)]

Accordingly,

AD/CD = CD/DB

The legs of the right-angled triangle are the mean proportion of the hypotenuse and the portion of the leg directly below the hypotenuse.

Mathematically, the leg rule says,

[Hypotenuse of Right Triangle]/[Legs of Right Triangle] = [Legs of Right Triangle]/[Part]

Accordingly,

AB/AC = AC/AD or AB/CB = CB/DB

1. Find the mean proportional between 4 and 9.

Solution:

Let the mean proportion between 4 and 9 is x.

Accordingly,

\[x^{2} = \sqrt{4 \times 9}\]

\[x^{2} = \sqrt{36}\]

x = 6

Therefore, the mean proportion between 4 and 9 is 6.

2. Find the mean proportional between 9 and 16.

Solution:

Let ‘k’ be the mean proportion between 9 and 16

Accordingly,

\[k^{2} = \sqrt{9 \times 16}\]

\[k^{2} = \sqrt{144}\]

k = 12

Therefore, the mean proportion between 9 and 16 is 12.

3. Find the value of x ( the length of AB)?

Solution:

Let us first find the length of the hypotenuse side BC.

BC = BD + DC = 16

Now, using the leg rule:

(Hypotenuse of Right Triangle)/(Legs of Right Triangle) = (Legs of Right Triangle)/(Part )

Substituting the value in the above leg rule formula, we get

16/x = x/9

\[x^{2} = \sqrt{16 \times 9}\]

\[x^{2} = \sqrt{144}\]

x = 12

4. What is the height (h) of the altitude AD?

Solution:

Using the altitude rule, we get

[One Part of Hypotenuse (Left)]/Altitude = Altitude/[Other Part of Hypotenuse (Right)]

Substituting the value in the above altitude rule equation, we get

4.9/x = x/10

\[x^{2} = \sqrt{4.9 \times 10}\]

\[x^{2} = \sqrt{49}\]

x = 7

In geometric mean or mean proportion, the values of both the ‘x’ are equal.

Mean Proportion is also referred to as Geometric Mean.

FAQ (Frequently Asked Questions)

1. Define Proportion.

Ans: A proportion is an expression that states two given ratios are equal. It can be written as two equal fractions p/q = q/r or using a colon, p:q = q:r. The following proportion is read as fifteen is to forty - five as three is to nine.

15/45 = 3/9.

2. How to Find a Mean Proportion?

Ans: There are different uses of the square root one of which is to find a mean proportion between any two numbers. Following are the steps to find a mean proportion between any two numbers:

Multiply the two given together.

Calculate square root out of their product, and it will be the mean proportion.

The resultant answer will be the mean proportion.

3. What Does the Geometric Mean Theorem State?

Ans: The geometric mean theorem describes the relationship between the length of the altitude on the hypotenuse in a right-angled triangle and two line segments it forms on the hypotenuse. It states that the geometric mean of two segments is equivalent to altitude.

If 'h' represents the altitude of right angles triangle and x and y are two segments on the hypotenuse, the altitude rule can be stated as:

h = √xy

h² = xy