Distance of a chord from the centre · 2 marks

In a circle, a chord is 5 cm away from the centre. If the radius of the circle is 13 cm, what is the length of the chord?

Answer: Half-chord = √(13² − 5²) = √144 = 12 cm, so the chord is 24 cm.

Step-by-step solution

Given: Radius OB = 13 cm; Distance OM = 5 cm (OM ⟂ AB)
To find: Length of chord AB

Idea: The perpendicular from the centre bisects the chord, so the radius, the distance and half the chord form a right triangle.

Let AB be the chord and OM ⟂ AB with OM = 5 cm. The perpendicular from the centre bisects the chord: AB = 2MB.½ mark
In right ΔOMB: MB² = OB² − OM² = 13² − 5² = 169 − 25 = 144, so MB = 12 cm.1 mark
AB = 2 × 12 = 24 cm.½ mark

The chord is 24 cm long.

Check: 5² + 12² = 25 + 144 = 169 = 13² ✓.

OB = 13 cm, OM = 5 cm, OM ⟂ AB

MB² = OB² − OM² = 169 − 25 = 144 (Baudhāyana–Pythagoras theorem)

MB = 12 cm

AB = 2MB (perpendicular from the centre bisects the chord)

∴ AB = 24 cm

Giving 12 cm, which is only half the chord.
Computing √(13² + 5²) instead of subtracting.
Forgetting to write the reason that the perpendicular from the centre bisects the chord.

MCQ (1 mark)

A chord of a circle of radius 7.3 cm is 4.8 cm from the centre. Its length is

Show answer

(B) 11 cm
Half-chord = √(7.3² − 4.8²) = √(53.29 − 23.04) = √30.25 = 5.5 cm, so the chord is 2 × 5.5 = 11 cm.

Find the length of a chord that is 15 cm from the centre of a circle of radius 25 cm.

Show answer

Half-chord = √(625 − 225) = √400 = 20 cm, so the chord is 40 cm.