Mean Proportional in Mathematics

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 refers 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 - \[\sqrt{pq}\]

Define Mean Proportion

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

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.

Mean Proportional Example

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.

Right Angles Mean Proportion

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.

Example:

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:

\[\frac{AB}{AC} = \frac{AC}{AD}, \frac{AB}{CB} = \frac{CB}{DB}, \frac{AD}{CD} = \frac{CD}{DB}\]

Altitude Rule

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,

\[\frac{\text{One Part of Hypotenuse(Left)}}{Altitude} = \frac{Altitude}{\text{Other Part of Hypotenuse(Right)}}\]

Accordingly,

\[\frac{AD}{CD} = \frac{CD}{DB}\]

Leg Rule

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,

\[\frac{\text{Hypotenuse Right Triangle}}{\text{Legs of Right Triangle}} = \frac{\text{Legs of Right Triangle}}{Part}\]

Accordingly,

\[\frac{AB}{AC} = \frac{AC}{AD} or \frac{AB}{CB} = \frac{CB}{DB}\]

Mean Proportion Formula

Mean Proportional Examples With Solutions

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:

\[\frac{\text{(Hypotenuse of Right Triangle)}}{\text{(Legs of Right Triangle)}} =  \frac{\text{(Legs of Right Triangle)}}{\text{(Part )}}\]

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

\[\frac{16}{x} =  \frac{x}{9}\]

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

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

x = 12

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

Solution:

Using the altitude rule, we get

\[\frac{\text{One Part of Hypotenuse (Left)}}{\text{Altitude}} =   \frac{\text{Altitude}}{\text{Other Part of Hypotenuse (Right)}}\]

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

\[\frac{4.9}{x} =  \frac{x}{10}\]

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

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

x = 7

Importance of Mean Proportional

Mean Proportion is one of the most essential topics of mathematics. It involves various theorems to find a relationship between two or more numbers. Mean Proportional is important for you for the following reasons. The basics of the Importance of Mean proportion will come in handy in subsequent topics that you will learn in your Mathematics class. With Mean Proportional, you can learn the different methods of finding altitude, Hypotenuse, and base of a Right-angled Triangle. The examples will give you an idea of how to solve different types of questions based on the concepts of Mean proportion.

By using the formulas of Mean proportion, you can also find out whether a set of given numbers is directly proportional or inversely. The Mean Proportional provides you with a plethora of examples to make you understand the concepts clearly. 

The Mean Proportional will become helpful study material during your revisions and exam preparations. It will give you a quick overview of the different concepts that come under Mean proportion. You can use Mean Proportional to revise the entire topic and solve questions related to it. Once you have thoroughly gone through the Mean Proportional, solving practical questions will be much easier.  

Solving questions related to Mean Proportional

Solving questions related to Mean Proportional will be quite easy if you know all the formulas and theorems from the chapter. You can go through the Mean Proportional on Vedantu’s website for absolutely free. Once you have learned all the formulas and their applications, you will be ready to solve questions based on Mean proportion. Use the following tips and tricks while solving such questions to become more efficient: 

• Before you start with the questions, make sure you have a clear understanding of all the topics that come under Mean proportion.

• Read the questions carefully to understand which formulas or theorems will be used in them to obtain the correct answers.

• Refer to the examples of Mean proportion to understand the Right sequence of steps you have to follow to solve a particular problem. 

• Make sure that you know the derivations of a formula or a theorem as they will come in handy while solving a complex question based on Mean proportion.

• Do not use any shortcuts to solve a question. Since there is a step-by-step marking in your maths exam, you should solve a question in the Right manner by mentioning every step. 

• Refer to the Mean Proportional whenever you are stuck on a question. 

• Go through all the formulas thoroughly so that you do not forget them while questions related to Mean Proportional. 

• Apart from the derivations and formulas, you should also know the definitions of each concept to understand its Meaning.    

Facts to Remember

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

• Mean Proportion is also referred to as Geometric Mean