Chapter 6.3, Problem 49STP

To determine

The coordinates of the fourth vertex of a parallelogram.

Answer to Problem 49STP

Option D. is correct.

Explanation of Solution

Given:

The vertices of the parallelogram are:

  $\left(0,0\right),\left(3,5\right),\left(0,5\right)$

Calculation:

The diagonals of a parallelogram bisect each other and both diagonals have the same midpoint.

Let the fourth vertex be (a, b).

So, the vertices will be:

  $A\left(0,0\right),B\left(a,b\right),C\left(3,5\right),D\left(0,5\right)$

The parallelogram formed by points A, B, C and D, will have the midpoints of AC and BD coinciding.

Therefore,

  $\begin{array}{l}AC\text{ midpoint = BD}\text{}\text{ midpoint}\\ \left(\frac{0+3}{2},\frac{0+5}{2}\right)=\left(\frac{0+a}{2},\frac{5+b}{2}\right)\\ \left(\frac{3}{2},\frac{5}{2}\right)=\left(\frac{0+a}{2},\frac{5+b}{2}\right)\\ \text{Equate the abscissa and ordinates:}\\ \frac{0+a}{2}=\frac{3}{2}\\ a=3\\ And,\\ \frac{5+b}{2}=\frac{5}{2}\\ b=0\\ So,\\ \text{Fourth coordinate is: }B\left(3,0\right)\end{array}$

Option D. is correct.