Chapter 10.1, Problem 20E

To determine

Find the distance between the given points and midpoint of the segment.

Answer to Problem 20E

The distance between the points is $\sqrt{4+9{\omega }^2}$ and the midpoint coordinates of the segment are $\left(\omega -1,\frac{5\omega }{2}\right)$ .

Explanation of Solution

Given:

Thegiven points are $\left(\omega -2,\omega \right)$ and $\left(\omega ,4\omega \right)$ .

Calculation:

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

If the coordinates of $p_1$ and $p_2$ are $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ , respectively, then the midpoint of $\overline{p_1{p}_2}$ has coordinates $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$ .

For the given points

  $\begin{array}{l}{x}_1=\omega -2,x_2=\omega \\ y_1=\omega ,y_2=4\omega \end{array}$

  $\begin{array}{l}d=\sqrt{{\left[\omega -\left(\omega -2\right)\right]}^2+{\left(4\omega -\omega \right)}^2}\\ =\sqrt{{\left(2\right)}^2+{\left(3\omega \right)}^2}\\ =\sqrt{4+9{\omega }^2}\\ \text{Midpoint}=\left(\frac{\omega -2+\omega }{2},\frac{\omega +4\omega }{2}\right)\\ =\left(\frac{2\omega -2}{2},\frac{5\omega }{2}\right)\\ =\left(\omega -1,\frac{5\omega }{2}\right)\end{array}$

Hence the distance between the points is $\sqrt{4+9{\omega }^2}$ and the midpoint coordinates of the segment are $\left(\omega -1,\frac{5\omega }{2}\right)$ .