91Ó°ÊÓ

Given that the quadrilateral has vertices$R\left(−1,−6\right),A\left(1,−3\right),Y\left(11,1\right)\text{and}J\left(9,−2\right)$

We have to discover and prove something about the quadrilateral.

The length of the quadrilateral = RA and YJ.

The breadth of the quadrilateral = AY and JR.

The diagonal of the quadrilateral = RY and AJ.

We will use the distance formula to find the lengths of .

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

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

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

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

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

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

We see that the pairs of opposite sides have equal length but the diagonals are of different lengths.

It means, the quadrilateral RAYJ is a parallelogram.