Chapter 13.4, Problem 8MRE

To determine

To find the coordinates of the fourth vertex of a rectangle.

Answer to Problem 8MRE

The coordinates of the fourth vertex of a rectangle is, $\left(-3,5\right)$ .

Explanation of Solution

Given information: The three vertices of a rectangle is $\left(-3,-2\right),\left(2,-2\right)and\left(2,5\right)$ .

Calculation: The required fourth vertex of a rectangle is obtained as,

Figure is,

  

Let fourth vertex is $\left(x,y\right)$ .

  $\because$ diagonal of rectangle bisects,

So, midpoints of both the diagonals will be same.

Midpoint of first diagonal,

  $\begin{array}{l}M=\left(\frac{2-3}{2},\frac{5-2}{2}\right)\\ =\left(-\frac{1}{2},\frac{3}{2}\right)\end{array}$

Midpoint of second diagonal,

  $M=\left(\frac{x+2}{2},\frac{y-2}{2}\right)$

Equating both,

  $\begin{array}{l}\frac{x+2}{2}=-\frac{1}{2}\\ \Rightarrow x+2=-1\\ \Rightarrow x=-3\end{array}$

  $\begin{array}{l}\frac{y-2}{2}=\frac{3}{2}\\ \Rightarrow y-2=3\\ \Rightarrow y=5\end{array}$

Hence, the coordinates of the fourth vertex of a rectangle is, $\left(-3,5\right)$ .