Chapter 1, Problem 1RE

For Exercises 1-2,

a. Find the exact distance between the points.

b. Find the midpoint of the line segment whose endpoints are the given points.

$\left(-1,8\right)\text{ and }\left(4,-2\right)$

To determine

To calculate: The exact distance between the given points.

Answer to Problem 1RE

The exact distance between the points $A\left(-1,8\right)$ and $B\left(4,-2\right)$ is $5\sqrt{5}$ .

Explanation of Solution

Calculation:

Consider two points $A\left(x_{1,}{y}_1\right)$ and $B\left(x_2,y_2\right)$ .Distance between two points is $d$ , which can be calculated as follows:

$\begin{array}{c}d=\sqrt{\left[{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2\right]}\\ =\sqrt{\left[{\left(4-\left(-1\right)\right)}^2+{\left(-2-8\right)}^2\right]}\\ =\sqrt{125}\\ =5\sqrt{5}\end{array}$

Hence, the exact distance between the given points is $5\sqrt{5}$ .

To determine

To calculate: The midpoint of the line segment whose endpoints are the given points.

Answer to Problem 1RE

The midpoint of the given endpoints $A\left(-1,8\right)$ and $B\left(4,-2\right)$ is $\left(\frac{3}{2},3\right)$ .

Explanation of Solution

Calculation:

Consider the endpoints of the given line segment as $A\left(x_{1,}{y}_1\right)$ and $B\left(x_2,y_2\right)$ .The midpoint of the given endpoints is $M\left(x_m,y_m\right)$ , which is calculated as follows:

$\begin{array}{c}M\left(x_m,y_m\right)=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ =\left(\frac{-1+4}{2},\frac{8+\left(-2\right)}{2}\right)\\ =\left(\frac{3}{2},3\right)\end{array}$

Hence, the midpoint of the given endpoints is $\left(\frac{3}{2},3\right)$ .