GATE CS 2025 Set 2 Q.9: Spatial Aptitude

In the given figure, PQRS is a square of side 2 cm and PLMN is a rectangle. The corner L of the rectangle is on the side QR. Side MN of the rectangle passes through the corner S of the square.

What is the area (in $\text{cm}^2$ ) of the rectangle PLMN?

Note: The figure shown is representative.

Square PQRS of side 2 cm with P at top-left, S at top-right, Q at bottom-left, and R at bottom-right; tilted rectangle PLMN has corner P, corner L on bottom side QR, and side MN passing through corner S.

$2\sqrt{2}$

$2$

$8$

$4$

Official answer and solution

Official answer: D
From: Official GATE 2025 answer key

Solution

Relate the area of both square PQRS and rectangle PLMN to the area of triangle PLS.

Consider $\triangle PLS$ inside both figures:

Relative to square $PQRS$ : Side $PS = 2\text{ cm}$ is the base of $\triangle PLS$ , and since vertex $L$ lies on the opposite parallel side $QR$ , the perpendicular height of $\triangle PLS$ onto $PS$ is $PQ = 2\text{ cm}$ . Therefore: $\text{Area}(\triangle PLS) = \frac{1}{2} \times PS \times PQ = \frac{1}{2} \times 2 \times 2 = 2\text{ cm}^2$
Relative to rectangle $PLMN$ : Side $PL$ is one side of the rectangle, and vertex $S$ lies on the opposite parallel side $MN$ . Thus the perpendicular distance from $S$ to $PL$ equals the other side length $PN$ of the rectangle, giving: $\text{Area}(\triangle PLS) = \frac{1}{2} \times PL \times PN = \frac{1}{2} \text{Area}(PLMN)$

Combining the two equations gives $\text{Area}(PLMN) = 2 \times \text{Area}(\triangle PLS) = 2 \times 2 = 4\text{ cm}^2$ .

Final answer

Option (D)

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2025 Computer Science and Information Technology - Session 2 (CS2)
Question: 9 of 65
Section: General Aptitude

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