Chapter 13.3, Problem 16WE

To determine

To find the type of quadrilateral HIJK is.

Answer to Problem 16WE

The points HIJK are Rhombus.

Explanation of Solution

Given information: The points are,

  $H\left(0,1\right)\text{, }I\left(2,-3\right)\text{, }J\left(-2,-1\right)\text{, }K\left(-4,3\right)$ .

Calculation: The required solutions is obtained as,

  $H\left(0,1\right)\text{, }I\left(2,-3\right)\text{, }J\left(-2,-1\right)\text{, }K\left(-4,3\right)$

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

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

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

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

All sides are congruent.

Now, find the slopes,

  $\begin{array}{l}{m}_{HI}=\frac{-3-1}{2-0}=-2\\ m_{IJ}=\frac{-1-\left(-3\right)}{-2-2}=-\frac{1}{2}\\ m_{JK}=\frac{3-\left(-1\right)}{-4-\left(-2\right)}=-2\\ m_{KH}=\frac{1-3}{0+4}=-\frac{1}{2}\end{array}$

Both pairs of opposite sides are parallel.

Hence, it is Rhombus.