A chord is drawn 12cm away from the centre of a circle of radius 13cm. What is the length of the chord?

Analyze the question: A chord is 12 cm from the centre of a circle whose radius is 13 cm. We need the length of the chord.

Concept: The perpendicular from the centre of a circle to a chord bisects the chord. This gives a right-angled triangle with:

• Hypotenuse = radius = 13 cm
• One side = perpendicular distance from the centre = 12 cm
• Other side = half the chord

Solution:

Let half the chord = x.

x² + 12² = 13² x² = 169 − 144 = 25 x = 5 cm

Full chord = 2 × 5 = 10 cm

(This is the 5-12-13 Pythagorean triple.)

Checking the options: The calculated length is 10.00 cm, which matches option E.

Common mistakes:

• Option C (25 cm) comes from forgetting to take the square root, or from adding 12 + 13.
• Options A and B are just 13² and 12², so they are not lengths of the chord.
• Stopping at 5 cm means you forgot to double it, since the perpendicular bisects the chord.

Tip: Whenever you see a chord and its distance from the centre, draw the perpendicular from the centre and use Pythagoras. Memorise the triples 3-4-5, 5-12-13, 8-15-17 and 7-24-25, because they save time in JAMB, WASSCE and NECO.