Q. 41 - Geometry

Question

Discover and prove something about the quadrilateral with vertices

R(-1, -6), A(1, -3), Y(11, 1), and J(9, -2)

Step-by-Step Solution

Answer

The quadrilateral RAYJ is a parallelogram.

Step-1 – Given

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

Step-2 – To determine

We have to discover and prove something about the quadrilateral.

Step-3 – Calculation

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.