Chapter 1, Problem 1RE

To determine

To find: The exact distance between points $A\left(-1,16\right)$ and $B\left(5,-8\right)$.

Answer to Problem 1RE

The exact distance between points $A\left(-1,16\right)$ and $B\left(5,-8\right)$ is $6\sqrt{17}\text{unit}$.

Explanation of Solution

Given:

The coordinates of points A and B are $\left(-1,16\right)$ and $\left(5,-8\right)$ respectively.

Formula used:

The distance $\left(d\right)$ between two points with coordinates $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is obtained by the distance formula $d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$.

Calculation:

Compare the points $A\left(-1,16\right)$ and $B\left(5,-8\right)$ with $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ respectively, and use the distance formula.

$\begin{array}{c}d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(5-\left(-1\right)\right)}^2+{\left(-8-16\right)}^2}\\ =\sqrt{{\left(5+1\right)}^2+{\left(-24\right)}^2}\\ =\sqrt{6^2+{\left(-24\right)}^2}\end{array}$

Simplify the above result further as below.

$\begin{array}{c}d=\sqrt{6^2+{\left(-24\right)}^2}\\ =\sqrt{6^2+{24}^2}\\ =\sqrt{6^2+{\left(6\cdot 4\right)}^2}\\ =\sqrt{6^2+6^2\cdot 4^2}\end{array}$

Solve the above result further as below.

$\begin{array}{c}d=\sqrt{6^2+6^2\cdot 4^2}\\ =\sqrt{6^2\left(1+4^2\right)}\\ =6\sqrt{\left(1+16\right)}\\ =6\sqrt{17}\end{array}$

Thus, the exact distance between points $A\left(-1,16\right)$ and $B\left(5,-8\right)$ is $6\sqrt{17}\text{unit}$.