In the given figure, PQRS is a quadrilateral. If QR = 18 cm and PS = 9 cm, then what is the area (in cm²) of quadrilateral PQRS?

Solution (By Examveda Team)

QP and RS extended to M to make an equilateral ΔMQR.
∠SPM = 180° - 150° = 30°
In ΔMPS

PS = 9 (Given)
$$\eqalign{ & \tan {30^ \circ } = \frac{{{\text{MS}}}}{{{\text{PS}}}} \cr & \frac{1}{{\sqrt 3 }} = \frac{{{\text{MS}}}}{9} \cr & {\text{MS}} = \frac{9}{{\sqrt 3 }} \cr & {\text{MS}} = 3\sqrt 3 \cr} $$
Area of quadrilateral PQRS = Area of Equilateral Δ (MQR) - Area of ΔMSP
$$\eqalign{ & = \frac{{\sqrt 3 }}{4} \times {\left( {18} \right)^2} - \frac{1}{2} \times 9 \times 3\sqrt 3 \cr & = \frac{{\sqrt 3 }}{4} \times 18 \times 18 - \frac{{27\sqrt 3 }}{2} \cr & = \frac{{27\sqrt 3 }}{2}\left[ {6 - 1} \right] \cr & = \frac{{135\sqrt 3 }}{2}{\text{ c}}{{\text{m}}^2} \cr} $$