Chapter B, Problem 2E

To determine

To find: The distance between the given points.

Answer to Problem 2E

The distance between the points is $2\sqrt{29}$.

Explanation of Solution

Given:

The points are $\left(1,-3\right),\left(5,7\right)$.

Formula used:

$\begin{array}{l}{\text{The distance between the points P}}_1\left(x_1,y_1\right){\text{ and P}}_2\left(x_2,y_2\right)\text{is}\\ \left|P_1{P}_2\right|=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\end{array}$

Calculation:

Let the given points be,

$\left(x_1,y_1\right)=\left(1,-3\right)\text{ and }\left(x_2,y_2\right)=\left(5,7\right)$.

Use the above mentioned formula to find the distance between the points

$\begin{array}{c}\left|P_1{P}_2\right|=\sqrt{{\left(5-1\right)}^2+{\left(7-\left(-3\right)\right)}^2}\\ =\sqrt{{\left(4\right)}^2+{\left(7+3\right)}^2}\\ =\sqrt{{\left(4\right)}^2+{\left(10\right)}^2}\\ =\sqrt{16+100}\end{array}$

On further simplification, the equation becomes,

$\begin{array}{c}\left|P_1{P}_2\right|=\sqrt{116}\\ =\sqrt{4\times 29}\\ =\sqrt{4}\times \sqrt{29}\\ =2\sqrt{29}\end{array}$

Thus, the equation for x for the given modulus function is $2\sqrt{29}$.