Question-9

trigonometry
geometry
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 cm²) of the rectangle \(PLMN\)?

A bit of angle chasing gives us the following:

Now, using a bit of trigonometry, we see the following:

\[ PL = \cfrac{2}{\sin(\theta)}, PN = 2 \sin(\theta) \]

Therefore, the area of the rectangle \(PLMN\) is \(4\).