Chapter 9, Problem 31RP

To determine

To show:The given quadrilateral QUAD , whose vertices are given, is a rectangle.

Explanation of Solution

Given information:

Coordinates of the vertices

Formula used:

Opposite sides are equal

Proof :

QU = $\sqrt{{\left(-1-4\right)}^2+{\left(-4-11\right)}^2}=\sqrt{25+225}=\sqrt{250}=5\sqrt{10}\text{ units}$

UA = $\sqrt{{\left(1-4\right)}^2+{\left(12-11\right)}^2}=\sqrt{9+1}=\sqrt{10}\text{ units}$

AD = $\sqrt{{\left(1+4\right)}^2+{\left(12+3\right)}^2}=\sqrt{25+225}=\sqrt{250}\text{=5}\sqrt{10}\text{ units}$

DU = $\sqrt{{\left(-1+4\right)}^2+{\left(-4+3\right)}^2}=\sqrt{9+1}=\sqrt{10}\text{ units}$

Now, since

  $QU=AD\text{ and }UA=DU$

Hence given quadrilateral is rectangle

Hence proved