CAT 2023 Slot 2 QA Q18 Geometry

Question 18 of the Quantitative Ability (QA) section, CAT 2023 Slot 2, with CAT's official answer and a worked solution.

A triangle is drawn with its vertices on the circle \(C\) such that one of its sides is a diameter of \(C\) and the other two sides have their lengths in the ratio \(a: b\). If the radius of the circle is \(r\), then the area of the triangle is

A
\(\frac{a b r^2}{2\left(a^2+b^2\right)}\)
B
\(\frac{a b r^2}{a^2+b^2}\)
C
\(\frac{4 a b r^2}{a^2+b^2}\)
D
\(\frac{2 a b r^2}{a^2+b^2}\)

Answer: (D) (2 a b r^2)/(a^2+b^2) Geometry MCQ

Solution

A triangle with a diameter as one side is right-angled, with hypotenuse \(2r\).
Legs \(ak\) and \(bk\): \(k^2(a^2+b^2) = 4r^2\), so \(k^2 = \frac{4r^2}{a^2+b^2}\).
Area \(= \frac12 \cdot ak \cdot bk = \frac{ab k^2}{2} = \frac{2ab r^2}{a^2+b^2}\).

Shortcut. Try \(a=b=1\): isosceles right triangle area is \(r^2\); only D gives \(r^2\).