CAT 2023 Slot 1 QA Q19 Geometry

Question 19 of the Quantitative Ability (QA) section, CAT 2023 Slot 1, with CAT's official answer and a worked solution.

In a right-angled triangle ABC, the altitude AB is 5 cm, and the base BC is 12 cm. P and Q are two points on BC such that the areas of ΔABP, ΔABQ and ΔABC are in arithmetic progression. If the area of ΔABC is 1.5 times the area of ΔABP, the length of PQ, in cm, is

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

Answer: 2 Geometry TITA

Solution

  1. Area \(ABC = \frac12\times5\times12 = 30\), so area \(ABP = \frac{30}{1.5} = 20\).
  2. AP of 20, \([ABQ]\), 30 gives \([ABQ] = 25\).
  3. All share height \(AB = 5\): \(BP = \frac{2\times20}{5} = 8\), \(BQ = \frac{2\times25}{5} = 10\).
  4. \(PQ = 10 - 8 = 2\) cm.

Shortcut. Areas are proportional to bases on BC: \(BP = 8\), \(BQ = 10\).

Same concept in other years

See every question on Triangles with a common height: areas proportional to bases.

More Geometry questions from CAT

All 65 Geometry questions asked since 2017.