Chapter 9.1, Problem 16WE

To determine

To calculate: The distance between the pairs of points.

Answer to Problem 16WE

The distance between the given points is $\sqrt{9\left(a^2+b^2\right)}$ .

Explanation of Solution

Given information: The coordinates of points are $\left(-a,b\right)$ and $\left(2a,4b\right)$ .

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

Calculation: Apply the distance formula to calculate the distance between the two points.

  $\begin{array}{cc}A& B\\ x_1=-a& x_2=2a\\ y_1=b& y_2=4b\end{array}$

  $\begin{array}{l}d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ d=\sqrt{{\left(2a-\left(-a\right)\right)}^2+{\left(4b-b\right)}^2}\\ d=\sqrt{{\left(3a\right)}^2+{\left(3b\right)}^2}\\ d=\sqrt{9a^2+9b^2}\\ d=\sqrt{9\left(a^2+b^2\right)}\end{array}$

Therefore, the required distance between the points is $\sqrt{9\left(a^2+b^2\right)}$ .

To determine

To calculate: The midpoint of line segment.

Answer to Problem 16WE

The coordinates of the midpoint of the line segment are $\left(\frac{a}{2},\frac{5b}{2}\right)$ .

Explanation of Solution

Given information: The coordinates of points are $\left(-a,b\right)$ and $\left(2a,4b\right)$ .

Formula used: $M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

Calculation: Since the line segment joining the given points, consider point M as a midpoint.

  $\begin{array}{cc}A& B\\ x_1=-a& x_2=2a\\ y_1=b& y_2=4b\end{array}$

  $\begin{array}{l}M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ M=\left(\frac{-a+2a}{2},\frac{b+4b}{2}\right)\\ M=\left(\frac{a}{2},\frac{5b}{2}\right)\end{array}$

Hence, the coordinates of the midpoint of the line segment are $\left(\frac{a}{2},\frac{5b}{2}\right)$ .