Question:
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
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
Updated On: Jul 29, 2025
- 8
- 6√2
- 5√3
- 2π
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The Correct Option is C
Solution and Explanation
To solve the problem of finding the length of a chord that subtends an angle of \(120^\circ\) at the center of the circle, given a chord of 5 cm that subtends an angle of \(60^\circ\), we use the formula for chord length:
\[ L = 2r \sin\left(\frac{\theta}{2}\right) \]
Step 1: Find the radius \( r \) using the given chord:
- Given: A chord of 5 cm subtends an angle of \(60^\circ\) at the center
- Using the formula: \( 5 = 2r \sin(30^\circ) \)
- Since \( \sin(30^\circ) = \frac{1}{2} \), we get: \( 5 = 2r \cdot \frac{1}{2} \)
- This simplifies to: \( 5 = r \)
Step 2: Use this radius to find the chord length for an angle of \(120^\circ\):
- Using the formula: \( L = 2r \sin(60^\circ) \)
- Since \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \), we get:
- \( L = 2 \cdot 5 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3} \)
Answer: The length of the chord that subtends a \(120^\circ\) angle at the center is \( \boxed{5\sqrt{3}} \) cm.
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