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. 8
• B. 6√2
• C. 5√3
• D. 2π

Answer 1

The Correct Option is C

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.