CAT 2021 Slot 3 QA Q11 Geometry

Question 11 of the Quantitative Ability (QA) section, CAT 2021 Slot 3, with CAT's official answer and a worked solution.

Let ABCD be a parallelogram. The lengths of the side AD and the diagonal AC are 10 cm and 20 cm, respectively. If the angle ∠ADC is equal to 30° then the area of the parallelogram, in sq. cm, is

A
\(\frac{25(\sqrt{3}+\sqrt{15})}{2}\)
B
\(25(\sqrt{5}+\sqrt{15})\)
C
\(\frac{25(\sqrt{5}+\sqrt{15})}{2}\)
D
\(25(\sqrt{3}+\sqrt{15})\)

Answer: (D) 25(√(3)+√(15)) Geometry MCQ

Solution

In triangle ADC: \(AD = 10\), \(AC = 20\), \(\angle D = 30^\circ\). Let \(DC = x\).
Cosine rule: \(400 = 100 + x^2 - 2(10)x\cos 30^\circ \Rightarrow x^2 - 10\sqrt3 x - 300 = 0\).
\(x = \frac{10\sqrt3 + \sqrt{300 + 1200}}{2} = 5\sqrt3 + 5\sqrt{15}\).
Area \(= AD \cdot DC \sin 30^\circ = 5x = 25(\sqrt3 + \sqrt{15})\).