Two parallel chords of lengths 10 cm and 24 cm are on the same side of the centre of a circle. The distance between the chords is 7 cm. Find the radius of the circle.
Step-by-step solution
To find: Radius r
Idea: The longer chord is nearer the centre. Write the radius twice, once with each chord (half-chord and distance), and set the two expressions equal.
- The longer chord (24 cm) is nearer the centre. Let its distance from O be x cm; then the 10 cm chord is (x + 7) cm from O (same side). The perpendicular from O bisects each chord: half-chords 12 cm and 5 cm.1 mark
- Radius from each chord: r2 = 122 + x2 and r2 = 52 + (x + 7)2.1 mark
- 144 + x2 = 25 + x2 + 14x + 49 ⇒ 14x = 70 ⇒ x = 5.1 mark
- r2 = 144 + 25 = 169 ⇒ r = 13 cm.1 mark
Check: Distances 5 and 12 differ by 7 ✓; 122 + 52 = 169 and 52 + 122 = 169 ✓.
Answer to write in the exam
Let OM ⟂ AB (24 cm), ON ⟂ CD (10 cm); OM = x, ON = x + 7 (same side)
AM = 12 cm, CN = 5 cm (perpendicular from the centre bisects the chord)
r2 = 122 + x2 = 52 + (x + 7)2
144 + x2 = 25 + x2 + 14x + 49
14x = 70 ⇒ x = 5
r2 = 144 + 25 = 169
∴ r = 13 cm
Common mistakes that cost marks
- Putting the 10 cm chord nearer the centre. The longer chord is always nearer.
- Using x and 7 − x: that setting is for chords on opposite sides.
- Using whole chords (24 and 10) instead of halves (12 and 5).
How this can come in the exam
In a circle of radius 20 cm, parallel chords of 24 cm and 32 cm lie on the same side of the centre. The distance between them is
- 4 cm
- 8 cm
- 28 cm
- 12 cm
Show answer
(A) 4 cm
Distances: √(400 − 144) = 16 cm and √(400 − 256) = 12 cm. Same side: 16 − 12 = 4 cm.
Try one yourself
Two parallel chords of lengths 112 cm and 126 cm are on the same side of the centre of a circle, 17 cm apart. Find the radius.
Show answer
Let the 126 cm chord be x from the centre: 632 + x2 = 562 + (x + 17)2 ⇒ 34x = 544 ⇒ x = 16, so r = √(3969 + 256) = 65 cm.
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