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 $latex a$ and $latex b$ is $latex \frac{a+b}{2}$, while their** geometric mean** is $latex \sqrt{ab}$. Now, we represent them using the figure below.

7 December 2013, Created with GeoGebra

Consider the semi-circle with diameter $latex \overline{AB}$ and radius $latex \overline{EF}$. We construct $latex \overline{CD} \perp \overline{AB}$ *anywhere* such that $latex C$ is on the circle and $latex D$ is on the diameter.If we let the length of $latex \overline{AB} = a$ and $latex \overline{BD} = b$, then the radius $latex \overline{EF} = (a+b)/2$.