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CAT 2017 Slot 1 — QA Question 17

Basics of TrianglesEasy

Let ABC be a right-angled isosceles triangle with hypotenuse BC. Let BQC be a semi-circle, away from A, with diameter BC. Let BPC be an arc of a circle centered at A and lying between BC and BQC. If AB has length 6 cm then the area, in sq. cm, of the region enclosed by BPC and BQC is:

Answer & solution

Correct answer: 18

  • A

    9π - 18

  • 18

  • C

    9π

  • D

    9

Solution

Easy

This is a classic "lune" result. Let AB=aAB=a. The semicircle on BCBC and the arc BPCBPC (quarter-circle centred at AA) both relate to aa in a way that makes the area between them collapse exactly to the triangle's area — no π\pi survives.

A B C Q P
1

Set the lengths. With legs AB=AC=a=6AB=AC=a=6 and right angle at AA:

BC=a2+a2=a2\begin{aligned} &BC=\sqrt{a^2+a^2}=a\sqrt{2} \end{aligned}
2

Area of semicircle BQCBQC. Its radius is BC2=a2\tfrac{BC}{2}=\tfrac{a}{\sqrt2}:

[semicircle BQC]=12π ⁣(a2)2=14πa2\begin{aligned} &[\text{semicircle }BQC]=\tfrac12\pi\!\left(\tfrac{a}{\sqrt2}\right)^2=\tfrac14\pi a^2 \end{aligned}
3

Area below arc BPCBPC. BPCBPC is a quarter-circle (radius aa, centre AA, 90∘90^\circ); the region between chord BCBC and arc BPCBPC is that quadrant minus △ABC\triangle ABC.

[quadrant A]=14πa2[between BC and BPC]=14πa2−[ABC]\begin{aligned} &[\text{quadrant }A]=\tfrac14\pi a^2\\ &[\text{between }BC\text{ and }BPC]=\tfrac14\pi a^2-[ABC] \end{aligned}
4

Subtract to get the enclosed region. Region between the two arcs = semicircle area −- the step-3 region:

Area=14πa2−(14πa2−[ABC])⇒ [ABC]=12⋅6⋅6=18(π cancels)\begin{aligned} &\text{Area}=\tfrac14\pi a^2-\left(\tfrac14\pi a^2-[ABC]\right)\\ &\Rightarrow\ [ABC]=\tfrac12\cdot 6\cdot 6=18 \quad\text{(}\pi\text{ cancels)} \end{aligned}
Area=18 sq cm\text{Area}=18\ \text{sq cm}

Recognise the lune: the area between the two arcs is always the triangle's area, so just compute 12⋅AB⋅AC=18\tfrac12\cdot AB\cdot AC = 18 directly.

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CAT 2017 Slot 1 QA Q17: Let ABC be a right-angled isosceles triangle with hypotenuse BC. Let BQC be a semi-circle, away from A, with d — Solution | TheCATExam