Chapter 9.1, Problem 15WE

To determine

To calculate: The distance between the pairs of points.

Answer to Problem 15WE

The distance between the given points is ‘a’.

Explanation of Solution

Given information: The coordinates of points are $\left(a,b\right)$ and $\left(0,b\right)$ .

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.

  $\begin{array}{cc}A& B\\ x_1=a& x_2=0\\ y_1=b& y_2=b\end{array}$

  $\begin{array}{l}d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ d=\sqrt{{\left(0-a\right)}^2+{\left(b-b\right)}^2}\\ d=\sqrt{a^2}\\ d=a\end{array}$

Therefore, the required distance between the points is ‘a’.

To determine

To calculate: The midpoint of line segment.

Answer to Problem 15WE

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

Explanation of Solution

Given information: The coordinates of points are $\left(a,b\right)$ and $\left(0,b\right)$ .

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.

  $\begin{array}{cc}A& B\\ x_1=a& x_2=0\\ y_1=b& y_2=b\end{array}$

  $\begin{array}{l}M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ M=\left(\frac{a+0}{2},\frac{b+b}{2}\right)\\ M=\left(\frac{a}{2},\frac{2b}{2}\right)\\ M=\left(\frac{a}{2},\frac{b}{2}\right)\end{array}$

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