AB is a chord of a circle with centre O. ON is drawn perpendicular from O to chord AB. If AB = 20 cm and ON = 2√11 cm, what is the radius (in cm) of the circle?

Introduction / Context:
This geometry question uses the relationship between a chord, the perpendicular distance from the centre to the chord and the radius of a circle. It is a direct application of the Pythagoras theorem to a right triangle formed by the radius, the perpendicular and half of the chord.

Given Data / Assumptions:

• Circle has centre O.
• AB is a chord of the circle with length AB = 20 cm.
• ON is perpendicular from O to AB, so ON ⟂ AB.
• ON = 2√11 cm.
• We must find the radius r = OA of the circle.
• Because ON is perpendicular to AB, it bisects the chord AB at N.

Concept / Approach:
When a perpendicular is drawn from the centre of a circle to a chord, it bisects the chord. Hence, AN = NB = AB / 2. Considering triangle ONA, we get a right triangle with hypotenuse OA (the radius), one leg ON and the other leg AN. Applying the Pythagoras theorem gives: OA^2 = ON^2 + AN^2 We can compute both ON^2 and AN^2 from the given information and then take the square root to find OA.

Step-by-Step Solution:
Step 1: Since ON ⟂ AB, chord AB is bisected at N. Step 2: Therefore, AN = NB = AB / 2 = 20 / 2 = 10 cm. Step 3: Consider right triangle ONA with right angle at N. Step 4: ON = 2√11 cm and AN = 10 cm. Step 5: Apply Pythagoras theorem: OA^2 = ON^2 + AN^2. Step 6: Compute ON^2: (2√11)^2 = 4 * 11 = 44. Step 7: Compute AN^2: 10^2 = 100. Step 8: So OA^2 = 44 + 100 = 144. Step 9: Taking the square root, OA = √144 = 12 cm.

Verification / Alternative check:
We can confirm the value by checking triangle ONA numerically. If OA = 12, ON = 2√11 ≈ 6.64 and AN = 10, then ON^2 + AN^2 ≈ 44 + 100 = 144, while OA^2 = 144. The equality holds, confirming that OA is indeed 12 cm. This is consistent with the geometry of a circle and a chord at a given distance from the centre.

Why Other Options Are Wrong:
10 cm would make the radius equal to half the chord, which is only possible if the chord is a diameter, in which case ON would be zero. 13 cm and 15 cm produce OA^2 values larger than 144 and do not satisfy OA^2 = ON^2 + AN^2 with the given lengths. 9 cm is smaller than half the chord length, which is impossible because the radius must be at least as long as the distance from the centre to any point on the chord plus the perpendicular distance from the centre to the chord does not exceed the radius.

Common Pitfalls:
A typical mistake is to forget that the perpendicular from the centre to a chord bisects the chord, leading to an incorrect value of AN. Another error is miscomputing (2√11)^2 or mishandling the addition of squares. Carefully identifying the right triangle and keeping the Pythagoras theorem in mind helps avoid these problems and makes such questions straightforward.

Final Answer:
The radius of the circle is 12 cm.