Question

If the length of a chord of a circle is equal to that of radius of the circle, then the angle subtended in radius, at the centre of the circle by the chord is

• A. 1
• B. \( \frac{\pi}{2} \)
• C. \( \frac{\pi}{3} \)
• D. \( \frac{\pi}{4} \)

Hint:

When the length of a chord is equal to the radius, the central angle is always \( \frac{\pi}{2} \).

Answer 1

The Correct Option is B

Step 1: Understanding the Geometry
Let the radius of the circle be \( r \), and let the length of the chord be \( r \) as well.
The central angle subtended by the chord can be determined using the cosine rule in the triangle formed by the radius lines and the chord.
Step 2: Apply the Cosine Rule
In the isosceles triangle formed by the two radii and the chord, the angle at the center \( \theta \) satisfies: \[ \cos \left( \frac{\theta}{2} \right) = \frac{\text{half of chord}}{r} = \frac{r/2}{r} = \frac{1}{2} \] Thus, \( \frac{\theta}{2} = \frac{\pi}{3} \), and hence \( \theta = \frac{\pi}{2} \).
Step 3: Conclusion
Thus, the angle subtended at the center of the circle by the chord is \( \frac{\pi}{2} \).