91Ó°ÊÓ

The given coordinates are $B\left(8,0\right)\text{,}T\left(2,−9\right)\text{,}R\left(−1,−7\right)\text{,and}C\left(5,2\right)$.

We have to find the area of the rectangle.

Using the distance formula, we will find the length and breadth of the rectangle.

BT and RC are the lengths of the rectangle.

TR and CB are the breadths of the rectangle.

So,

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

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

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

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

We know that in a rectangle the pair of opposite sides are equal. So, BTRC is a rectangle.

Area of the rectangle is:

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

So, the area of the rectangle is 39.