CAT 2018

Slot 2

QA

Question & Solution

1. Concept Used

• Topic: Circles — Chord Length and Central Angle
• Formula: $$ \text{Chord Length} = 2r \cdot \sin\left(\frac{\theta}{2}\right) $$

where $r$ is the radius of the circle and $\theta$ is the central angle subtended by the chord.

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2. Calculation

Step 1: Find the radius using the first chord.

We are given that a chord of length $5$ cm subtends a central angle of $60°$. Using the chord-length formula:

$$5 = 2r \cdot \sin\left(\frac{60°}{2}\right) = 2r \cdot \sin(30°) = 2r \cdot \frac{1}{2} = r$$

So the radius $r = 5$ cm.

Step 2: Find the length of the second chord.

Now we need the length of a chord that subtends a central angle of $120°$ at the centre of the same circle (radius $= 5$ cm):

$$\text{Chord Length} = 2r \cdot \sin\left(\frac{120°}{2}\right) = 2 \times 5 \times \sin(60°)$$

$$= 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3} \text{ cm}$$

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3. Solution

Answer = Option A ✅

The length of the chord that subtends an angle of $120°$ at the centre is $\mathbf{5\sqrt{3}}$ cm.