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

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

Given Data / Assumptions:

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

Concept / Approach:
When a perpendicular is drawn from the centre of a circle to a chord, it bisects the chord. Thus each half of AB is 10 cm. In the right triangle formed by the radius, half the chord and the perpendicular from the centre, we can apply Pythagoras theorem: r^2 = (half chord)^2 + (distance from centre)^2 So: r^2 = 10^2 + (2√11)^2

Step-by-Step Solution:
Step 1: Since ON is perpendicular to AB at N, it bisects AB. Step 2: Therefore, AN = NB = AB / 2 = 20 / 2 = 10 cm. Step 3: Consider right triangle ONA, where OA is the radius r, ON = 2√11 cm and AN = 10 cm. Step 4: Apply Pythagoras theorem: OA^2 = ON^2 + AN^2. Step 5: Compute ON^2: (2√11)^2 = 4 * 11 = 44. Step 6: Compute AN^2: 10^2 = 100. Step 7: So r^2 = OA^2 = 44 + 100 = 144. Step 8: Therefore, r = √144 = 12 cm.

Verification / Alternative check:
We can check that a radius of 12 cm is reasonable. The distance from the centre to the chord is 2√11 ≈ 6.64 cm, and half the chord is 10 cm. Together with the radius 12 cm, these values form a valid right triangle since 6.64^2 + 10^2 ≈ 144, confirming the calculation.

Why Other Options Are Wrong:
10 cm: This would imply the radius equals half the chord, which is only true if the chord is a diameter. Here, ON is not zero, so AB is not a diameter.
13 cm and 15 cm: These are larger than necessary and do not satisfy the equation r^2 = 44 + 100 exactly.
9 cm: This is too small; with such a radius, half chord 10 cm would exceed the radius, which is impossible in a circle.

Common Pitfalls:
A common error is forgetting that the perpendicular from the centre to the chord bisects the chord, leading to incorrect half chord length. Another mistake is miscomputing (2√11)^2 or misapplying Pythagoras theorem. Keeping the right triangle picture in mind and carefully squaring the radical avoids these mistakes.

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