Chapter 13.3, Problem 18WE

To determine

To find the type of quadrilateral HIJK is.

Answer to Problem 18WE

The points HIJK are Parallelogram.

Explanation of Solution

Given information: The points are,

  $H\left(-3,-3\right)\text{, }I\left(-5,-6\right)\text{, }J\left(4,-5\right)\text{, }K\left(6,-2\right)$ .

Calculation: The required solutions is obtained as,

  $H\left(-3,-3\right)\text{, }I\left(-5,-6\right)\text{, }J\left(4,-5\right)\text{, }K\left(6,-2\right)$

Distance of $HI=\sqrt{{\left(-3+5\right)}^2+{\left(-3+6\right)}^2}=\sqrt{13}$

Distance of $IJ=\sqrt{{\left(4-\left(-5\right)\right)}^2+{\left(-5-\left(-6\right)\right)}^2}=\sqrt{82}$

Distance of $JK=\sqrt{{\left(4-6\right)}^2+{\left(-5+2\right)}^2}=\sqrt{13}$

Distance of $KH=\sqrt{{\left(6-\left(-3\right)\right)}^2+{\left(-2+3\right)}^2}=\sqrt{82}$

Now, find the slopes,

  $\begin{array}{l}{m}_{HI}=\frac{-6+3}{-5+3}=\frac{3}{2}\\ m_{IJ}=\frac{-5+6}{4+5}=\frac{1}{9}\\ m_{JK}=\frac{-5+2}{4-6}=\frac{3}{2}\\ m_{KH}=\frac{-3+2}{-3-6}=\frac{1}{9}\end{array}$

Hence, it is parallelogram.