GATE CS 2025 Set 2 Q9 — General Aptitude

Let $\angle LPQ = \theta$. Since $PQRS$ is a square, $\angle PQR = 90^\circ$.

In the right-angled triangle $\triangle PQL$:

Since $PLMN$ is a rectangle, the opposite sides $PL$ and $MN$ are parallel. The width of the rectangle, $PN$, is the perpendicular distance between these two parallel lines.

The line $MN$ passes through the corner $S$. Since $\angle SPQ = 90^\circ$, the angle between $PS$ and $PL$ is $90^\circ - \theta$. Therefore, the perpendicular distance from $P$ to the line $MN$ is:

The area of the rectangle $PLMN$ is given by: