From the centre of a circle, a perpendicular of length 8 cm is drawn to a chord of length 12 cm. What is the diameter of the circle in centimetres?

Introduction / Context: This geometry question uses properties of chords in a circle. When a radius or line from the centre is drawn perpendicular to a chord, it bisects the chord. This fact, combined with Pythagoras theorem, allows us to determine the radius and hence the diameter of the circle from the chord length and the perpendicular distance. Given Data / Assumptions:

• Length of the chord AB is 12 cm.
• Perpendicular from the centre O to the chord has length OM = 8 cm.
• M is the midpoint of chord AB because OM is perpendicular from the centre.
• We need the diameter, which is 2 times the radius.

Concept / Approach: Key concepts:

• A perpendicular from the centre of a circle to a chord bisects the chord.
• In right triangle OMA, OA is the radius, OM is one leg and AM is half of the chord.
• Use Pythagoras theorem: OA² = OM² + AM².
• Once the radius is found, multiply by 2 to get the diameter.

Step-by-Step Solution: Step 1: Let AB be the chord and O be the centre of the circle. Step 2: Since OM is perpendicular to AB, point M is the midpoint of AB and AM = MB = 12 / 2 = 6 cm. Step 3: Consider the right triangle OMA with OM = 8 cm and AM = 6 cm. Step 4: Let the radius R = OA. Apply Pythagoras theorem: R² = OM² + AM². Step 5: Compute R² = 8² + 6² = 64 + 36 = 100. Step 6: Therefore R = √100 = 10 cm. Step 7: The diameter D = 2R = 2 × 10 = 20 cm. Verification / Alternative check: Step 1: If the diameter is 20 cm, the radius is 10 cm. Recompute the perpendicular distance using OM² = R² − AM². Step 2: OM² = 10² − 6² = 100 − 36 = 64, so OM = 8 cm, which matches the given perpendicular length. Step 3: This confirms that the radius and hence the diameter are correctly calculated. Why Other Options Are Wrong: Option 10 cm is the radius, not the diameter, and misses the final step. Options 12 cm, 16 cm and 18 cm do not satisfy Pythagoras theorem with legs 8 cm and 6 cm when taken as diameters. For example, a diameter of 16 cm would give a radius of 8 cm, where R² = 64, which is less than OM² + AM² = 100. Common Pitfalls: Some learners forget that the perpendicular from the centre bisects the chord and mistakenly use the full chord length instead of half. Another frequent mistake is to stop after computing the radius and not converting it to the diameter as required. Incorrect application of Pythagoras theorem, such as using R = OM − AM, leads to wrong results. Final Answer: The diameter of the circle is 20 cm.