Chapter 6.5, Problem 5CYU

To determine

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

Answer to Problem 5CYU

The quadrilateral is a rhombus.

Explanation of Solution

Given information :

First we shall find the distance of all the sides.

  $\begin{array}{l}QR=\sqrt{{\left(-1-2\right)}^2+{\left(-2-1\right)}^2}=\sqrt{18}\\ RS=\sqrt{{\left(-4-\left(-1\right)\right)}^2+{\left(1-\left(-2\right)\right)}^2}=\sqrt{18}\\ ST=\sqrt{{\left(-1-\left(-4\right)\right)}^2+{\left(4-1\right)}^2}=\sqrt{18}\\ TQ=\sqrt{{\left(-1-2\right)}^2+{\left(4-1\right)}^2}=\sqrt{18}\end{array}$

All sides are equal.

Now we shall find the slope,

Slope of QS is $\frac{-4-2}{1-1}=\infty$

Slope of RT is $\frac{-1-\left(-1\right)}{4-\left(-2\right)}=0$

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

Therefore the quadrilateral is a rhombus.