Chapter 10, Problem 1RE

To determine

To calculate: The length and midpoint of the line segment between the points $\left(0,0\right)$ and $\left(5,-12\right)$.

Answer to Problem 1RE

Solution:

The length of the line segment between the points $\left(0,0\right)$ and $\left(5,-12\right)$ is $\underset{\_}{13\text{ units}}$ and the midpoint is $\left(\frac{5}{2},-6\right)$.

Explanation of Solution

Given information:

The points are $\left(0,0\right)$ and $\left(5,-12\right)$.

Formula used:

The distance formula between the two points $(x_1,y_1)\text{ and }(x_2,y_2)$,

$d=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$

The midpoint of any two points $(x_1,y_1)\text{ and }(x_2,y_2)$,

$\text{Midpoint}=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

Calculation:

Consider the points $\left(0,0\right)\text{ and }(5,-12)$.

The distance between the two points $(x_1,y_1)\text{ and }(x_2,y_2)$ is,

$d=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$

Substitute 0 for $x_1$, 5 for $x_2$, $-12$ for $y_1$ and 0 for $y_2$.

$\begin{array}{c}d=\sqrt{{(5-0)}^2+{(-12-0)}^2}\\ =\sqrt{25+144}\\ =\sqrt{169}\\ =13\text{ units}\end{array}$

The midpoint between the line segment of two points $(x_1,y_1)\text{ and }(x_2,y_2)$ is,

$\text{Midpoint}=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

Substitute 0 for $x_1$, 5 for $x_2$, $-12$ for $y_1$ and 0 for $y_2$.

$\begin{array}{c}\text{Midpoint}=\left(\frac{0+5}{2},\frac{0+(-12)}{2}\right)\\ =\left(\frac{5}{2},-6\right)\end{array}$

Hence, the length of the line segment between the points $\left(0,0\right)\text{ and }(5,-12)$ is 13 units and the midpoint of the line segment is $\left(\frac{5}{2},-6\right)$.