Chapter 5.2, Problem 28PSC

To determine

To Prove: The diagonals of quadrilateral PQRS bisect each other..

Explanation of Solution

Given:

  

Concept Used:

Parallelogram is a quadrilateral of which opposite sides are parallel and equal in length.

  $\text{Coordinates of mid-point of side having end-points}\left(x_1,y_1\right)\text{and}\left(x_2,y_2\right)=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

Proof:

As we know that

  $\text{Coordinates of mid-point of side having end-points}\left(x_1,y_1\right)\text{and}\left(x_2,y_2\right)=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

Coordinates of mid-points of PQRS are

  $\begin{array}{l}\text{Coordinates of mid-point of side }\overline{PQ}=\left(\frac{0+0}{2},\frac{0+2b}{2}\right)=\left(0,b\right)\\ \text{Coordinates of mid-point of side }\overline{QR}=\left(\frac{0+2a}{2},\frac{2b+2b}{2}\right)=\left(a,2b\right)\\ \text{Coordinates of mid-point of side }\overline{RS}=\left(\frac{2a+2a}{2},\frac{2b+0}{2}\right)=\left(2a,b\right)\\ \text{Coordinates of mid-point of side }\overline{PS}=\left(\frac{0+2a}{2},\frac{0+0}{2}\right)=\left(a,0\right)\\ \text{Now, mid-point of side }\overline{PR}=\left(\frac{0+2a}{2},\frac{2b+0}{2}\right)=\left(a,b\right)\\ \text{Mid-point of side }\overline{QS}=\left(\frac{0+2a}{2},\frac{2b+0}{2}\right)=\left(a,b\right)\\ \Rightarrow \text{Diagonals of quadrilateral PQRS bisect each other}\text{.}\end{array}$