NinetynineCAT 2023 Slot 2 QA Q17 Geometry
Question 17 of the Quantitative Ability (QA) section, CAT 2023 Slot 2, with CAT's official answer and a worked solution.
In a rectangle ABCD, AB = 9 cm and BC = 6 cm. P and Q are two points on BC such that the areas of the figures ABP, APQ, and AQCD are in geometric progression. If the area of the figure AQCD is four times the area of triangle ABP, then BP : PQ : QC is
- A1 : 1 : 2
- B1 : 2 : 4
- C2 : 4 : 1
- D1 : 2 : 1
Answer: (C) 2 : 4 : 1 Geometry MCQ
Solution
- Triangles ABP and APQ both have height \(AB = 9\): areas \(4.5\,BP\) and \(4.5\,PQ\). Rectangle area \(= 54\).
- Three terms in GP with third \(= 4 \times\) first, so the ratio is 2. Hence APQ \(= 2\,\)ABP, so \(PQ = 2BP\), and AQCD \(= 4 \times 4.5BP = 18BP\).
- Sum: \(4.5BP + 9BP + 18BP = 31.5BP = 54 \Rightarrow BP = \frac{12}{7}\), \(PQ = \frac{24}{7}\), \(QC = 6 - \frac{36}{7} = \frac{6}{7}\).
- \(BP : PQ : QC = 12 : 24 : 6 = 2 : 4 : 1\).