CAT 2022 Slot 1 QA Q11 Geometry

Question 11 of the Quantitative Ability (QA) section, CAT 2022 Slot 1, with CAT's official answer and a worked solution.

A trapezium \(\text{ABCD}\) has side \(\text{AD}\) parallel to \(\text{BC}, \angle \text{BAD}=90^{\circ}, \text{BC}=3 \text{~cm}\) and \(\text{AD}=8 \text{~cm}\). If the perimeter of this trapezium is \(36 \text{~cm}\), then its area, in sq. \(\text{cm}\), is

TITA: type in the answer. There are no options and no negative marking.

Answer: 66 Geometry TITA

Solution

AB is perpendicular to both parallel sides, so AB is the height \(h\). Dropping a perpendicular from C to AD, the slant side \(CD = \sqrt{h^2 + (8-3)^2} = \sqrt{h^2 + 25}\).
Perimeter: \(h + 3 + 8 + \sqrt{h^2+25} = 36 \Rightarrow \sqrt{h^2 + 25} = 25 - h\).
Squaring: \(h^2 + 25 = 625 - 50h + h^2 \Rightarrow h = 12\).
Area \(= \frac{1}{2}(3 + 8)(12) = 66\).

Shortcut. 5-12-13 triangle: \(h + CD = 25\) with difference 5 gives \(h = 12, CD = 13\).