Chapter 6.5, Problem 4CYP

To determine

To determine if the given vertices are of a rhombus, a rectangle or a square.

Answer to Problem 4CYP

The quadrilateral is a rhombus.

Explanation of Solution

Given information :

First we shall find the distance of all the sides.

  $\begin{array}{l}JK=\sqrt{{\left(-11-0\right)}^2+{\left(8-5\right)}^2}=\sqrt{120}\\ KL=\sqrt{{\left(-14-\left(-11\right)\right)}^2+{\left(-3-8\right)}^2}=\sqrt{120}\\ LM=\sqrt{{\left(-3-\left(-14\right)\right)}^2+{\left(-6-\left(-3\right)\right)}^2}=\sqrt{120}\\ MJ=\sqrt{{\left(-3-0\right)}^2+{\left(-6-5\right)}^2}=\sqrt{120}\end{array}$

All sides are equal.

Now we shall find the slope,

Slope of JL is $\frac{-14-0}{-3-5}=\frac{7}{4}$

Slope of KM is $\frac{-3-\left(-11\right)}{-6-8}=\frac{-4}{7}$

When the product of slope is -1, then the lines are perpendicular. Here lines are perpendicular

Therefore the quadrilateral is a rhombus.