Question

A chord of a circle of diameter 24 cm subtends an angle $60^\circ$ at the centre of the circle. Then, the length of the chord, in cm, is

• A. $12\sqrt{3}$
• B. 18
• C. 12
• D. 24

Hint:

Whenever a chord subtends an angle of $60^\circ$ at the center of a circle, the triangle formed by the chord and the two radii becomes an equilateral triangle. Therefore, the length of the chord is equal to the radius of the circle. Alternatively, remember the formula \[ \text{Chord Length} = 2r\sin\left(\frac{\theta}{2}\right). \] For this question, \[ 2(12)\sin30^\circ = 12 \text{ cm}. \]

Answer 1

The Correct Option is C

Concept: When a chord of a circle subtends an angle at the center, the two radii joining the center to the endpoints of the chord form an isosceles triangle. If the central angle is $60^\circ$, then the triangle formed by the two radii and the chord becomes an equilateral triangle because all three angles become equal to $60^\circ$. Another standard formula for the length of a chord is: \[ \text{Chord Length} = 2r\sin\left(\frac{\theta}{2}\right) \] where
• $r$ is the radius of the circle,
• $\theta$ is the angle subtended at the center. Both methods lead to the same answer.

Step 1: Determine the radius of the circle. The diameter of the circle is given as \[ 24 \text{ cm}. \] Since radius is half the diameter, \[ r=\frac{24}{2}=12 \text{ cm}. \] Thus, \[ OA=OB=12 \text{ cm}, \] where $O$ is the center of the circle and $A$ and $B$ are the endpoints of the chord.

Step 2: Form the triangle using the chord and the radii. Join the center $O$ to the endpoints of the chord. Then triangle $AOB$ is formed with \[ OA=OB=12 \text{ cm} \] and \[ \angle AOB=60^\circ. \] Since $OA=OB$, triangle $AOB$ is an isosceles triangle. The sum of the angles of a triangle is $180^\circ$. Therefore, \[ \angle OAB+\angle OBA+60^\circ=180^\circ. \] Since \[ \angle OAB=\angle OBA, \] we get \[ 2\angle OAB=120^\circ. \] Hence, \[ \angle OAB=60^\circ. \] Similarly, \[ \angle OBA=60^\circ. \] Thus, \[ \angle AOB = \angle OAB = \angle OBA = 60^\circ. \] Therefore, triangle $AOB$ is an equilateral triangle.

Step 3: Determine the length of the chord. Since triangle $AOB$ is equilateral, \[ AB=OA=OB. \] But \[ OA=12 \text{ cm}. \] Therefore, \[ AB=12 \text{ cm}. \] Hence, the length of the chord is \[ \boxed{12 \text{ cm}}. \] Alternative Verification Using the Chord Formula The chord length formula is \[ AB = 2r\sin\left(\frac{\theta}{2}\right). \] Substituting \[ r=12 \] and \[ \theta=60^\circ, \] we get \[ AB = 2(12)\sin 30^\circ. \] Since \[ \sin 30^\circ=\frac{1}{2}, \] \[ AB = 24\times\frac{1}{2} = 12 \text{ cm}. \] This confirms the result obtained earlier. Final Answer: \[ \boxed{12 \text{ cm}} \] Therefore, the correct option is \[ \boxed{\text{(C) 12}}. \]