Chapter 13.1, Problem 41WE

To determine

To discover and prove something about the quadrilateral.

Explanation of Solution

Given information:

The quadrilateral has vertices

  $R\left(-1,-6\right),A\left(1,-3\right),Y\left(11,1\right)\text{ and }J\left(9,-2\right)$

Proof:

The quadrilateral can be discover as

Use distance formula to find the value of $RA,AY,YJ,JR,RY\text{ and }AJ$

RA and YJ is the length of the quadrilateral

AY and JR is the breadth of the quadrilateral

RY and AJ is the diagonal of the quadrilateral

Therefore,

  $\begin{array}{l}RA=\sqrt{{\left(1-\left(-1\right)\right)}^2+{\left(-3-\left(-6\right)\right)}^2}\left[\text{since, }R\left(-1,-6\right)\text{ and }A\left(1,-3\right)\right]\\ RA=\sqrt{{\left(1+1\right)}^2+{\left(-3+6\right)}^2}\\ RA=\sqrt{{\left(2\right)}^2+{\left(3\right)}^2}\\ RA=\sqrt{4+9}\\ RA=\sqrt{13}\end{array}$

  $\begin{array}{l}AY=\sqrt{{\left(11-1\right)}^2+{\left(1-\left(-3\right)\right)}^2}\left[\text{since, }A\left(1,-3\right)\text{ and }Y\left(11,1\right)\right]\\ AY=\sqrt{{\left(11-1\right)}^2+{\left(1+3\right)}^2}\\ AY=\sqrt{{\left(10\right)}^2+{\left(4\right)}^2}\\ AY=\sqrt{100+16}\\ AY=\sqrt{116}\end{array}$

  $\begin{array}{l}YJ=\sqrt{{\left(9-11\right)}^2+{\left(-2-1\right)}^2}\left[\text{since, }Y\left(11,1\right)\text{ and }J\left(9,-2\right)\right]\\ YJ=\sqrt{{\left(-2\right)}^2+{\left(-3\right)}^2}\\ YJ=\sqrt{4+9}\\ YJ=\sqrt{13}\end{array}$

  $\begin{array}{l}JR=\sqrt{{\left(-1-9\right)}^2+{\left(-6-\left(-2\right)\right)}^2}\left[\text{since, }J\left(9,-2\right)\text{ and }R\left(-1,-6\right)\right]\\ JR=\sqrt{{\left(-1-9\right)}^2+{\left(-6+2\right)}^2}\\ JR=\sqrt{{\left(-10\right)}^2+{\left(-4\right)}^2}\\ JR=\sqrt{100+16}\\ JR=\sqrt{116}\end{array}$

  $\begin{array}{l}RY=\sqrt{{\left(11-\left(-1\right)\right)}^2+{\left(1-\left(-6\right)\right)}^2}\left[\text{since, }R\left(-1,-6\right)\text{ and }Y\left(11,1\right)\right]\\ RY=\sqrt{{\left(11+1\right)}^2+{\left(1+6\right)}^2}\\ RY=\sqrt{{\left(12\right)}^2+{\left(7\right)}^2}\\ RY=\sqrt{144+49}\\ RY=\sqrt{193}\end{array}$

  $\begin{array}{l}AJ=\sqrt{{\left(9-1\right)}^2+{\left(-2-\left(-3\right)\right)}^2}\left[\text{since, }A\left(1,-3\right)\text{ and }J\left(9,-2\right)\right]\\ AJ=\sqrt{{\left(9-1\right)}^2+{\left(-2+3\right)}^2}\\ AJ=\sqrt{{\left(8\right)}^2+{\left(1\right)}^2}\\ AJ=\sqrt{64+1}\\ AJ=\sqrt{65}\end{array}$

Since, pairs of opposite sides are equal but the length of the diagonals are different

So, RAYJ is a parallelogram

Hence

The quadrilateral RAYJ is a parallelogram.