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Area of Right Triangle with Inscribed Circle

CAT 2021 Slot 2 · Quantitative Ability · Easy · Geometry

This is an easy Quantitative Ability question from the CAT 2021 Slot 2 paper. It tests Geometry. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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A circle of diameter 8 inches is inscribed in a triangle ABC where ∠ABC=90∘\angle ABC = 90^\circ. If BC=10 inchesBC = 10\text{ inches} then the area of the triangle in square inches is

Answer

120

Explanation

Inradius r=diameter2=4r = \frac{\text{diameter}}{2} = 4. For a right-angled triangle with legs a=BC=10a = BC = 10 and b=ABb = AB, and hypotenuse c=ACc = AC: r=a+b−c2  ⟹  4=10+b−c2  ⟹  c−b=2r = \frac{a + b - c}{2} \implies 4 = \frac{10 + b - c}{2} \implies c - b = 2

Also, by Pythagoras theorem: c2−b2=a2=100  ⟹  (c−b)(c+b)=100c^2 - b^2 = a^2 = 100 \implies (c - b)(c + b) = 100 Since c−b=2c - b = 2, we get c+b=50c + b = 50. Adding c−b=2c - b = 2 and c+b=50c + b = 50 gives 2b=48  ⟹  b=242b = 48 \implies b = 24.

Area of right triangle ΔABC=12×a×b=12×10×24=120 square inches\Delta ABC = \frac{1}{2} \times a \times b = \frac{1}{2} \times 10 \times 24 = 120\text{ square inches}.

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