Chapter 1.8, Problem 45E

To determine

To calculate: The coordinates of the point S so that quadrilateral PQRS is a parallelogram.

Answer to Problem 45E

The coordinates of point S so that PQRS forms a parallelogram is $S\left(2,-3\right)$ .

Explanation of Solution

Given information:

The points $P\left(-1,-4\right),Q\left(1,1\right)\text{ and }R\left(4,2\right)$ .

Formula used:

A parallelogram is a quadrilateral in which diagonals bisect each other at the same point.

Mid-point formula between two points $\text{X}\left(a_1,b_1\right)$ and $\text{Y}\left(a_2,b_2\right)$ is mathematically expressed as,

  $\left(\frac{a_1+a_2}{2},\frac{b_1+b_2}{2}\right)$

Calculation:

Consider the provided vertices $P\left(-1,-4\right),Q\left(1,1\right)\text{ and }R\left(4,2\right)$ .

By plotting the given points on the coordinate plane, we get the following figure,

  

Recall that a parallelogram is a quadrilateral in which diagonals bisect each other at the same point.

So, to find the coordinates of S such that PQRS forms a parallelogram, the mid-point of its diagonals i.e. PR and QS must be equal.

Let the coordinates of the point be $\left(a,b\right)$ .

Recall that the mid-point formula between two points $\text{X}\left(a_1,b_1\right)$ and $\text{Y}\left(a_2,b_2\right)$ is mathematically expressed as,

  $\left(\frac{a_1+a_2}{2},\frac{b_1+b_2}{2}\right)$

So, midpoint of PR is calculated as,

  $\begin{array}{c}\left(\frac{-1+4}{2},\frac{-4+2}{2}\right)=\left(\frac{3}{2},\frac{-2}{2}\right)\\ =\left(\frac{3}{2},-1\right)\end{array}$

Now, midpoint of QS is calculated as,

  $\left(\frac{1+a}{2},\frac{1+b}{2}\right)$

Since, diagonals of a parallelogram bisect each other at same point, so midpoint of PR and QS is same.

So, equate the mid-points of PR and QS as,

  $\left(\frac{1+a}{2},\frac{1+b}{2}\right)=\left(\frac{3}{2},-1\right)$

Now, equate x-coordinate and y-coordinate from both sides of the equation as,

  $\begin{array}{c}\frac{1+a}{2}=\frac{3}{2}\\ 1+a=3\\ a=3-1\\ a=2\end{array}$

  $\begin{array}{c}\frac{1+b}{2}=-1\\ 1+b=-2\\ b=-2-1\\ b=-3\end{array}$

Thus, the coordinates of point S so that PQRS forms a parallelogram is $S\left(2,-3\right)$ .