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.
We will construct the geometric mean of a and b geometrically. So, here are the steps:
- Let’s start by drawing the diameter of a circle.
- Suppose the two endpoints are X and Y of the diameter.
- Choose any point M on this particular diameter
- Now, let’s construct a perpendicular at M which hits the circle at N.
- M divides the diameter into two parts XM and YM. Suppose, the length of XM and YM be a and b respectively.
- 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.
- 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:
- Since, we know that similar triangles have proportional sides. So, we can say,
Hence, this is proved.
Hope you liked it!
Some Useful Links:
- AM-GM Inequality – Math Olympiad Concepts Video
- Triangular Number Sequence – Explanation with Application