Chapter P, Problem 8T

To determine

To calculate: The distance between the points $\left(-\text{2,5}\right)\text{,}\left(\text{6,0}\right)$ and mid- point of the line segment joining these points by first plotting the points $\left(-\text{2,5}\right)\text{,}\left(\text{6,0}\right)$ in the plane.

The distance between the points $\left(-\text{2,5}\right)\text{,}\left(\text{6,0}\right)$ is $\sqrt{89}$ units and the mid-points of the line segment joining the point is $\left(2,\frac{5}{2}\right)$.

Calculation:

Consider the provided points $\left(-\text{2,5}\right)\text{,}\left(\text{6,0}\right)$.

Join the points and the graph as shown below,

The graph will be a straight line.

Consider the provided points $\left(-\text{2,5}\right)\text{,}\left(\text{6,0}\right)$.

If $\left(x_1,y_1\right),\left(x_2,y_1\right)$ are the points, then the distance formula is,

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

Use the above formula to find the distance for the points $\left(-\text{2,5}\right)\text{,}\left(\text{6,0}\right)$.

$\begin{array}{c}d=\sqrt{{\left(6-\left(-2\right)\right)}^2+{\left(0-5\right)}^2}\\ =\sqrt{{\left(8\right)}^2+{\left(-5\right)}^2}\\ =\sqrt{64+25}\\ =\sqrt{89}\end{array}$

Therefore, the distance is $\sqrt{89}\text{ units}$.

Now, to calculate the mid-point of the line segment joining the points $\left(-\text{2,5}\right)\text{,}\left(\text{6,0}\right)$.

If $\left(x_1,y_1\right)\text{ and }\left(x_2,y_2\right)$ are points, the mid-point $\left(x,y\right)$ is defined as,

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

Now, apply the above formula for points $\left(-\text{2,5}\right)\text{,}\left(\text{6,0}\right)$,

$\begin{array}{c}\left(x,y\right)=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ =\left(\frac{-2+6}{2},\frac{5+0}{2}\right)\\ =\left(\frac{4}{2},\frac{5}{2}\right)\\ =\left(2,\frac{5}{2}\right)\end{array}$

Thus, the mid-point of the points $\left(-\text{2,5}\right)\text{,}\left(\text{6,0}\right)$ is $\left(2,\frac{5}{2}\right)$.

Hence, the distance between the points $\left(-\text{2,5}\right)\text{,}\left(\text{6,0}\right)$ is $\sqrt{89}\text{ units}$ and the mid-point of the line segment joining the points is $\left(2,\frac{5}{2}\right)$.