Have you ever wondered what is the relationship between the arithmetic mean and the geometric mean of two numbers? Or you have probably heard some theorems about their relationship. In this post, we are going to see that their relationship is really very easy to imagine if represented geometrically.
The arithmetic mean of two numbers and is , while their geometric mean is . Now, we represent them using the figure below.
7 December 2013, Created with GeoGebra
Consider the semi-circle with diameter and radius . We construct anywhere such that is on the circle and is on the diameter.If we let the length of and , then the radius .
Now, move by dragging in the figure above. What do you observe? What is the relationship between and ?
Yes, you are right. If we drag on the diameter of the semi-circle,
We already know that . We only need to get .
Notice that and are similar triangles (Prove it!). This means that
which gives us .
Now, substituting this to *, we have
which is the Arithmetic Mean Geometric Mean Inequality.
So, the Arithmetic Mean Geometric Mean Inequality states that the Geometric Mean of two numbers is always less than or equal to their Arithmetic Mean. Geometrically speaking, the segment perpendicular to the diameter that you can draw from the diameter of a semicircle to a point on the circle is shorter or the same length or of the same length as the radius.