Chapter A.9, Problem 122AYU

To determine

To show: That the arithmetic mean of $a$ and $b$ is equidistant from $a$ and $b$.

Answer to Problem 122AYU

The arithmetic mean $\frac{a + b}{2}$ is equidistant from $a$ and $b$.

Explanation of Solution

Let the arithmetic mean is $\frac{a + b}{2}$

The distance between two points on a number line is |a − b| or |b − a| 

From a to $\frac{a + b}{2}$, the distance is: $\left|a-\frac{a + b}{2}\right|$

From $\frac{a + b}{2}$ to b, the distance is: $\left|\left(a + b\right)/2-b\right|$

Set them equal to each other and simply and see if they equal each other.

Therefore the arithmetic mean $\frac{a + b}{2}$ is equidistant from $a$ and $b$.