CAT 2024 Slot 1 QA Q19 Geometry

Question 19 of the Quantitative Ability (QA) section, CAT 2024 Slot 1, with CAT's official answer and a worked solution.

ABCD is a rectangle with sides AB = 56 cm and BC = 45 cm, and E is the midpoint of side CD. Then, the length, in cm, of radius of incircle of \(\triangle A D E\) is

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

Answer: 10 Geometry TITA

Solution

Triangle \(ADE\) has a right angle at \(D\): \(AD = 45\), \(DE = \frac{56}{2} = 28\).
Hypotenuse \(AE = \sqrt{45^2 + 28^2} = \sqrt{2809} = 53\).
For a right triangle, inradius \(r = \frac{\text{leg}_1 + \text{leg}_2 - \text{hypotenuse}}{2} = \frac{45 + 28 - 53}{2} = 10\).

Shortcut. 28–45–53 is a Pythagorean triple; \(r = \frac{a + b - c}{2}\) for right triangles.