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August 11, 2020

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Let's learn how to calculate the geometric mean. This is a concept video useful for Mathematics Olympiad and ISI and CMI Entrance.

Watch and Learn:

Read and Learn: What is the Geometric mean of two numbers a and b & how to calculate it?

Suppose a and b are positive numbers then their geometric mean is defined as square root of a times b. This is the formula of geometric mean.

formula to calculate geometric mean

We will construct the geometric mean of a and b geometrically. So, here are the steps:

  1. Let's start by drawing the diameter of a circle.
  2. Suppose the two endpoints are X and Y of the diameter.
  3. Choose any point M on this particular diameter
  4. Now, let's construct a perpendicular at M which hits the circle at N.
  5. M divides the diameter into two parts XM and YM. Suppose, the length of XM and YM be a and b respectively.
  6. Now, we want to show that the length of MN is the square root of a times b, i.e., the geometric mean of a and b.
geometric mean of a and b

7. Now, join XN and YN, and then look at the triangles YMN and XMN and try to show that these two triangles are actually similar. Now, see the following proof:

Similarity proof of two triangles

8. Since, we know that similar triangles have proportional sides. So, we can say,

geometric mean proof

Hence, this is proved.

Hope you liked it!

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