A chord is 2cm from the centre of a circle. If the radius of the circle is 5cm, find the length of the chord

Analyze the question: A chord lies 2 cm from the centre of a circle of radius 5 cm. We need the chord's length.

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

• Hypotenuse = radius = 5 cm
• One side = distance from the centre to the chord = 2 cm
• Other side = half the chord

Solution:

Let half the chord = x. By Pythagoras' theorem:

x² + 2² = 5²

x² = 25 − 4 = 21

x = √21 cm

The full chord is twice this:

Chord = 2 × √21 = 2√21 cm

Answer: Option A.

Note that option D (√21 cm) is only half the chord. This is the trap in the question, so remember to double the value you get from Pythagoras.

Tip: In chord questions, always ask yourself whether you have found the half-chord or the whole chord. Draw the perpendicular from the centre, solve for the half-chord, and then multiply by 2.