Chapter 6.4, Problem 4CYP

To determine

To calculate: To determine whether JKLM is a rectangle using the Slope formula

Answer to Problem 4CYP

Quadrilateral JKLM is a not a rectangle

Explanation of Solution

Given: Quadrilateral JKLM has vertices $J\left(-10,2\right),K\left(-8,-6\right),L\left(5,-3\right),M\left(2,5\right)$

Formula Used:

Slope of line joining two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is calculated as follows:

  $Slope=\frac{y_2-y_1}{x_2-x_1}$

When slope of two lines are equal, then the lines are parallel to each other

When product of slope of two lines is equal to $-1$ , then the lines are perpendicular to each other.

Quadrilateral is a parallelogram if opposite sides are parallel to each other

Parallelogram is a rectangle is adjacent sides are perpendicular to each other

Calculation:

Givena Quadrilateral JKLM has vertices $J\left(-10,2\right),K\left(-8,-6\right),L\left(5,-3\right),M\left(2,5\right)$

Slope of sides of quadrilateral are calculated as follows:

  $m_{JK}=\frac{-6-2}{-8-\left(-10\right)}=\frac{-8}{-8+10}=\frac{-8}{2}=-4$

  $m_{KL}=\frac{-3-\left(-6\right)}{5-\left(-8\right)}=\frac{-3+6}{5+8}=\frac{3}{13}$

  $m_{LM}=\frac{5-\left(-3\right)}{2-5}=\frac{5+3}{-2}=\frac{8}{-2}=-4$

  $m_{MJ}=\frac{2-5}{-10-2}=\frac{-3}{-12}=\frac{1}{4}$

Since the sides KL and MJ are not parallel to each other, thus quadrilateral is not a rectangle

Conclusion:

Hence, quadrilateral JKLM is a not a rectangle