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CAT 2018 Slot 2 — QA Question 16

Basics of CirclesEasy

A chord of length 5 cm subtends an angle of 60° at the centre of a circle. The length, in cm, of a chord that subtends an angle of 120° at the centre of the same circle is

Answer & solution

Correct answer: 5 3

  • 53

  • B

    62

  • C

    2π

  • D

    8

Solution

Easy

The chord and the two radii form an isosceles triangle whose apex angle is the central angle. Use the 60∘60^\circ chord to find the radius, then apply the chord-length formula 2rsin⁡(θ/2)2r\sin(\theta/2) for the 120∘120^\circ chord.

1

Find the radius from the 60∘60^\circ chord. With apex angle 60∘60^\circ and two equal radii, the triangle is equilateral, so the chord equals the radius.

chord=5⇒r=5\begin{aligned} &\text{chord}=5\Rightarrow r=5 \end{aligned}
2

Chord length for central angle θ\theta. Dropping a perpendicular from the centre bisects both the chord and the angle, giving the standard formula.

chord=2rsin⁡ ⁣(θ2)\begin{aligned} &\text{chord}=2r\sin\!\left(\frac{\theta}{2}\right) \end{aligned}
3

Evaluate at θ=120∘\theta=120^\circ. Here θ/2=60∘\theta/2=60^\circ and sin⁡60∘=32\sin 60^\circ=\tfrac{\sqrt3}{2}.

chord=2⋅5⋅sin⁡60∘=10⋅32=53\begin{aligned} &\text{chord}=2\cdot 5\cdot\sin 60^\circ=10\cdot\frac{\sqrt3}{2}=5\sqrt3 \end{aligned}
O r = 5 5√3 120°
length=53 cm\text{length}=5\sqrt3\text{ cm}

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CAT 2018 Slot 2 QA Q16 — Basics of Circles solution | TheCATExam