Chapter 13.1, Problem 31WE

To determine

To find the area of the rectangle

Answer to Problem 31WE

The area of a rectangle is 39

Explanation of Solution

Given information:

The given points is

  $B\left(8,0\right)\text{ , }T\left(2,-9\right)\text{ , }R\left(-1,-7\right)\text{, and }C\left(5,2\right)$

Calculation:

The area of the rectangle can be obtained as

Use the distance formula to find the length and breadth of the rectangle

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

First find BT, TR, RC and CB

BT and RCis the length of the rectangle

TR and CB is the breadth of the rectangle

Therefore,

  $\begin{array}{l}BT=\sqrt{{\left(2-8\right)}^2+{\left(-9-0\right)}^2}\left[\text{since }B\left(8,0\right)\text{ and }T\left(2,-9\right)\right]\\ BT=\sqrt{{\left(-6\right)}^2+{\left(-9\right)}^2}\\ BT=\sqrt{36+81}\\ BT=\sqrt{117}\end{array}$

  $\begin{array}{l}TR=\sqrt{{\left(-1-2\right)}^2+{\left(-7-\left(-9\right)\right)}^2}\left[\text{since }T\left(2,-9\right)\text{ and }R\left(-1,-7\right)\right]\\ TR=\sqrt{{\left(-1-2\right)}^2+{\left(-7+9\right)}^2}\\ TR=\sqrt{{\left(-3\right)}^2+{\left(2\right)}^2}\\ TR=\sqrt{9+4}\\ TR=\sqrt{13}\end{array}$

  $\begin{array}{l}RC=\sqrt{{\left(5-\left(-1\right)\right)}^2+{\left(2-\left(-7\right)\right)}^2}\left[\text{since }R\left(-1,-7\right)\text{ and }C\left(5,2\right)\right]\\ RC=\sqrt{{\left(5+1\right)}^2+{\left(2+7\right)}^2}\\ RC=\sqrt{{\left(6\right)}^2+{\left(9\right)}^2}\\ RC=\sqrt{36+81}\\ RC=\sqrt{117}\end{array}$

  $\begin{array}{l}CB=\sqrt{{\left(8-5\right)}^2+{\left(0-2\right)}^2}\left[\text{since }C\left(5,2\right)\text{ and }B\left(8,0\right)\right]\\ CB=\sqrt{{\left(3\right)}^2+{\left(-2\right)}^2}\\ CB=\sqrt{9+4}\\ CB=\sqrt{13}\end{array}$

Pairs of opposite sites are equal so, BTRC is a rectangle

Area of rectangle

  $\begin{array}{l}a=l\times b\left[\begin{array}{l}a=\text{area}\\ l=\text{length}\\ b=\text{breadth}\end{array}\right]\\ a=\left(BT\right)\left(BC\right)\\ a=\left(\sqrt{117}\right)\left(\sqrt{13}\right)\\ a=\sqrt{1521}\\ a=39\end{array}$

Hence,

The area of a rectangle is 39