A chord of length 60 cm is at a perpendicular distance of 16 cm from the centre of a circle. What is the radius (in cm) of the circle?

Introduction / Context:
This question checks knowledge of the relationship between the radius of a circle, the length of a chord and the perpendicular distance from the centre to that chord. It uses a simple right triangle inside the circle. This is a standard mensuration and geometry fact that often appears in aptitude and school exams.

Given Data / Assumptions:

• Length of the chord is 60 cm.
• Perpendicular distance from the centre of the circle to the chord is 16 cm.
• We need to find the radius of the circle.
• The distance from the centre to the chord is along the perpendicular dropped from the centre.
• The circle is assumed to be perfect and the chord lies fully inside it.

Concept / Approach:
The perpendicular from the centre of a circle to a chord bisects the chord. Thus, the chord is split into two equal segments. These segments together with the radius form right triangles. If half the chord length is a, the perpendicular distance is d and the radius is R, then R^2 = a^2 + d^2 by Pythagoras theorem. We use this relation to find R. This approach is direct and avoids any complicated geometry.

Step-by-Step Solution:
Step 1: The chord length is 60 cm, so half of it is 60 / 2 = 30 cm. Step 2: Let half chord be a = 30 cm and perpendicular distance from centre to chord be d = 16 cm. Step 3: In the right triangle formed by radius, half chord and perpendicular, we have R^2 = a^2 + d^2. Step 4: Compute a^2 = 30^2 = 900 and d^2 = 16^2 = 256. Step 5: Add them to get R^2 = 900 + 256 = 1156. Step 6: Therefore R = √1156 = 34 cm.

Verification / Alternative check:
We can verify by reconstructing the chord length from the radius and distance. Using R = 34 cm and perpendicular distance d = 16 cm, half chord a = √(R^2 − d^2) = √(1156 − 256) = √900 = 30 cm. Doubling this gives chord length 60 cm, which matches the problem statement. Therefore the computed radius is fully consistent with all given measurements.

Why Other Options Are Wrong:
If R were 17 cm, 51 cm or 68 cm, then R^2 − d^2 would not give a perfect square corresponding to half chord 30 cm. For example, with R = 30 cm, R^2 − d^2 = 900 − 256 = 644, giving a half chord that does not match 30 cm. Hence all such values contradict the given chord length and distance. Only R = 34 cm ensures the right triangle condition and the correct chord length.

Common Pitfalls:
A common error is to forget that the perpendicular from the centre to a chord bisects it, leading to incorrect use of the full chord length instead of half. Another mistake is misapplying Pythagoras theorem, for example using R^2 = d^2 − a^2, which would not make sense in this context. Carefully drawing the diagram and labelling the right triangle helps keep track of which sides are legs and which one is the hypotenuse.

Final Answer:
The radius of the circle is 34 cm.