GATE 2022 EC GA Q.8: solution and answer

Official answer: CSource: Official answer key, GATE 2022 EC (IIT Kharagpur)

1. Plot the points. $P(0,1)$ and $Q(0,-3)$ both lie on the $y$-axis ($x=0$). $R(-2,-1)$ and $S(2,-1)$ both lie on the line $y=-1$. $R$ and $S$ are on opposite sides of $x=0$, and $P$ and $Q$ are on opposite sides of $y=-1$, so segments $PQ$ and $RS$ cross each other. Two crossing segments cannot both be sides of a quadrilateral, so $PQ$ and $RS$ are its diagonals and the boundary runs $P\to S\to Q\to R\to P$. (The letters are not given in boundary order.)

2. Diagonal lengths (distance between two points on the same vertical or horizontal line): $PQ=|1-(-3)|=4$ and $RS=|2-(-2)|=4$.

3. $PQ$ is vertical and $RS$ is horizontal, so the diagonals are perpendicular. Both have midpoint $(0,-1)$, so they bisect each other.

4. Diagonals that are equal, perpendicular and bisect each other make the quadrilateral a square (a rhombus with equal diagonals).

5. Area of a quadrilateral with perpendicular diagonals: $$\text{Area}=\frac{1}{2}{d}_1{d}_2=\frac{1}{2}\times 4\times 4=8$$

6. Check using a side: $PS=\sqrt{(2-0{)}^2+(-1-1{)}^2}=\sqrt{8}=2\sqrt{2}$, so the area of the square is $(2\sqrt{2}{)}^2=8$. This matches option (C).