Chapter 1.8, Problem 119E

To determine

To graph: The parallelogram PQRS where coordinates of the point P, Q and R are $P\left(0,3\right),Q\left(2,2\right),R\left(5,3\right)$ . Also compute the coordinate of the point Q .

Explanation of Solution

Given information:

PQRS is a parallelogram and coordinates of three points are $P\left(0,3\right),Q\left(2,2\right),R\left(5,3\right)$ .

Graph:

Plot the points $P\left(0,3\right),Q\left(2,2\right),R\left(5,3\right)$ on the coordinate plane.

  

Let the coordinates of the point S be $\left(x,y\right)$ .

It is known that diagonals of the parallelogram bisect each.

The parallelogram has two diagonals $PR$ and $QS$ .

Let $M\left(a,b\right)$ be the midpoint of the diagonal joined by the line segment $PR$ .

Recall that the midpoint of the segment that joins two points $A=\left(x_1,y_1\right)\text{ and }B=\left(x_2,y_2\right)$ in the Cartesian plane is denoted by $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$ .

Now, $M\left(a,b\right)$ is the mid-point of the line segment joined by the points $P\left(0,3\right)\text{ and }R\left(5,3\right)$ .

Therefore,

  $\begin{array}{c}\left(\frac{0+5}{2},\frac{3+3}{2}\right)=\left(a,b\right)\\ \left(\frac{5}{2},\frac{6}{2}\right)=\left(a,b\right)\\ \left(\frac{5}{2},3\right)=\left(a,b\right)\end{array}$

Therefore, $M\left(\frac{5}{2},3\right)$ is the mid-point of the line segment joined by the points $P\left(0,3\right)\text{ and }R\left(5,3\right)$ .

Also, $M\left(\frac{5}{2},3\right)$ is the mid-point of the line segment joined by the points $Q\left(2,2\right)\text{ and }S\left(x,y\right)$ .

  $\left(\frac{2+x}{2},\frac{2+y}{2}\right)=\left(\frac{5}{2},3\right)$

Equate the x -coordinate of each bracket,

  $\begin{array}{c}\frac{2+x}{2}=\frac{5}{2}\\ 2+x=5\\ x=3\end{array}$

Equate the y -coordinate of each bracket,

  $\begin{array}{c}\frac{2+y}{2}=3\\ 2+y=6\\ y=4\end{array}$

Therefore, coordinates of the point S are $S\left(3,4\right)$ .

Now, plot the points $P\left(0,3\right),Q\left(2,2\right),R\left(5,3\right)\text{ and }S\left(3,4\right)$ on the coordinate plane.

  

Interpretation:

The parallelogram PQRS so formed above, has diagonals that bisect each other. Opposites sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary that is adds to $180°$ .