Chapter 4, Problem 9CT

To determine

To find: whether the quadrilateral is a parallelogram or rectangle

Answer to Problem 9CT

The quadrilateral is a parallelogram

Explanation of Solution

Given:

  $J(1,7),K(6,4),L(2,-6),M(-3,-3)$

Calculation:

Consider the vertices of a quadrilateral

  $J(1,7),K(6,4),L(2,-6),M(-3,-3)$

Find the slope $m1$ for the line joining vertices $J(1,7)$ and $K(6,4)$

  $\begin{array}{l}m1=\frac{4-7}{6-1}\\ =-\frac{3}{5}\end{array}$

Find the slope $m2$ for the line joining vertices $K(6,4)$ and $L(2,-6)$

  $\begin{array}{l}m2=\frac{-6-4}{2-6}\\ =\frac{-10}{-4}\\ =\frac{5}{2}\end{array}$

Find the slope $m3$ for the line joining vertices $L(2,-6)$ and $M(-3,-3)$

  $\begin{array}{l}m3=\frac{-3-(-6)}{-3-2}\\ =-\frac{3}{5}\end{array}$

And, find the slope $m3$ for the line joining vertices $J(1,7)$ and $M(-3,-3)$

  $\begin{array}{l}m4=\frac{-3-7}{-3-1}\\ =\frac{-10}{-4}\\ =\frac{5}{2}\end{array}$

As the slope $m2,m4$ are same and $m1,m3$ are same

So, the opposite sides of the quadrilaterals are parallel which matches the condition for the parallelogram

Therefore, the quadrilateral is a parallelogram

Conclusion:

The quadrilateral is a parallelogram