Chapter 6.2, Problem 37PPS

To determine

The coordinates of vertex D.

Answer to Problem 37PPS

The coordinates of point D are (1, -1).

Explanation of Solution

Given:

ABCD is a parallelogram with vertices:

  $A\left(-2,5\right),B\left(2,2\right),C\left(4,-4\right).$

Calculation:

The diagonals of a parallelogram bisect each other.

Both of the diagonals have the same mid-point.

Therefore, the mid-points of AC and BD coincide.

Hence,

  $\begin{array}{l}\text{AC midpoint = BD midpoint}\\ \left(\frac{-1+3}{2},\frac{5-4}{2}\right)=\left(\frac{1+a}{2},\frac{2+b}{2}\right)\\ \left(\frac{2}{2},\frac{1}{2}\right)=\left(\frac{1+a}{2},\frac{2+b}{2}\right)\end{array}$

Equate the abscissa and the ordinates:

  $\begin{array}{l}\frac{1+a}{2}=\frac{2}{2}\\ 1+a=2\\ a=1\\ And;\\ \frac{2+b}{2}=\frac{1}{2}\\ b=1-2\\ b=-1\end{array}$

Therefore, the coordinates of point D are (1, -1).