In a circle, the length of a chord is equal to the radius of the circle. What is the angle, in radians, subtended at the centre of the circle by this chord?

Introduction / Context:
This question uses the relationship between a chord length, the radius of a circle and the central angle that subtends the chord. It is a standard trigonometric application in circle geometry and often appears in questions involving radians and arc lengths.

Given Data / Assumptions:

• A circle has radius R.
• A chord of the circle has length equal to R.
• The chord subtends some central angle θ at the centre of the circle.
• We are asked to find θ in radians.
• We can use the formula for chord length in terms of radius and central angle.

Concept / Approach:
For a circle of radius R, the length L of a chord that subtends a central angle θ (in radians) is: L = 2R * sin(θ / 2) Here, we are given that L = R. Substituting and simplifying gives an equation involving sin(θ / 2), which we can solve to find θ. Since we want the principal (smaller) central angle associated with that chord, we will choose the acute value that fits the geometry of the circle.

Step-by-Step Solution:
Step 1: Use the chord length formula L = 2R * sin(θ / 2). Step 2: Given L = R, set R = 2R * sin(θ / 2). Step 3: Divide both sides by R (R is non zero): 1 = 2 * sin(θ / 2). Step 4: Solve for sin(θ / 2): sin(θ / 2) = 1 / 2. Step 5: The principal angle whose sine is 1 / 2 is π / 6 (30°). Step 6: Therefore θ / 2 = π / 6, so θ = 2 * (π / 6) = π / 3. Step 7: Thus, the central angle subtended by the chord is π / 3 radians.

Verification / Alternative check:
You can cross check by substituting θ = π / 3 back into the chord formula. Then θ / 2 = π / 6, so sin(θ / 2) = sin(π / 6) = 1 / 2. The chord length becomes L = 2R * (1 / 2) = R, which matches the given condition. This confirms that π / 3 is the correct central angle for a chord whose length equals the radius.

Why Other Options Are Wrong:
1 radian is approximately 57.3°, and substituting θ = 1 into the formula does not give sin(θ / 2) = 1 / 2. π / 2 corresponds to 90°, which would give a chord longer than the radius (it would be √2 times the radius). π / 4 (45°) gives sin(π / 8), which is not 1 / 2, so that chord is shorter than R. 2π / 3 (120°) would correspond to sin(60°) = √3 / 2, giving a chord longer than the radius. Only π / 3 gives exactly sin(θ / 2) = 1 / 2 and hence chord length equal to the radius.

Common Pitfalls:
Some students mistakenly use the arc length formula s = Rθ instead of the chord length formula, which relates to the curved distance along the circle, not the straight line segment. Another pitfall is forgetting to divide the angle by 2 when using sin(θ / 2) in the chord formula. Carefully distinguishing between arc and chord and remembering the correct half angle relationship is crucial for solving this type of problem correctly.

Final Answer:
The central angle subtended by the chord is π / 3 radians.