Chapter 9.1, Problem 12WE

To determine

To calculate: The distance between the pairs of points.

Answer to Problem 12WE

The distance between the given points is $4.89$ .

Explanation of Solution

Given information: The coordinates of points are $\left(5,\sqrt{5}\right)$ and $\left(3,-\sqrt{5}\right)$ .

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.

  $\begin{array}{cc}A& B\\ x_1=5& x_2=3\\ y_1=\sqrt{5}& y_2=-\sqrt{5}\end{array}$

  $\begin{array}{l}d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ d=\sqrt{{\left(3-5\right)}^2+{\left(-\sqrt{5}-\sqrt{5}\right)}^2}\\ d=\sqrt{4+{\left(-2\sqrt{5}\right)}^2}\\ d=\sqrt{4+20}\\ d=4.89\end{array}$

Therefore, the required distance between the points is $4.89$ .

To determine

To calculate: The midpoint of line segment.

Answer to Problem 12WE

The coordinates of the midpoint of the line segment are $\left(4,0\right)$ .

Explanation of Solution

Given information: The coordinates of points are $\left(5,\sqrt{5}\right)$ and $\left(3,-\sqrt{5}\right)$ .

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.

  $\begin{array}{cc}A& B\\ x_1=5& x_2=3\\ y_1=\sqrt{5}& y_2=-\sqrt{5}\end{array}$

  $\begin{array}{l}M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ M=\left(\frac{5+3}{2},\frac{\sqrt{5}-\sqrt{5}}{2}\right)\\ M=\left(4,0\right)\end{array}$

Hence, the coordinates of the midpoint of the line segment are $\left(4,0\right)$ .