Circles • Topic 1 of 4

Chords & Arcs

Two facts unlock almost every chord question. First, the perpendicular dropped from the centre to a chord bisects it — so the centre O, the foot of the perpendicular M, and an endpoint A form a right triangle with legs d (distance from centre) and half-chord, and hypotenuse r. That gives the workhorse formula chord = 2√(r² − d²). Second, equal chords are equidistant from the centre, and conversely chords equidistant from the centre are equal — so the longer chord is always closer to the centre, and the diameter (distance 0) is the longest chord. In CAT, chord problems almost always reduce to a right triangle: draw the radius to an endpoint and the perpendicular to the chord, then use Pythagoras. Equal arcs subtend equal chords and equal central angles, which lets you transfer information around the circle quickly.

✅ Solved examples

1. A chord of length 16 cm lies in a circle of radius 10 cm. How far is the chord from the centre?
Half-chord = 8. d = √(r² − 8²) = √(100 − 64) = √36 = 6 cm.
2. A chord is 8 cm from the centre of a circle of radius 17 cm. Find its length.
Half = √(17² − 8²) = √(289 − 64) = √225 = 15. Chord = 2 × 15 = 30 cm.
3. Two parallel chords of a circle of radius 13 cm are 5 cm and 12 cm from the centre, on opposite sides. Find the distance between them.
Chords are on opposite sides, so distance = 5 + 12 = 17 cm.
4. Two equal chords of a circle intersect inside it. Show they make equal distances to the centre, and if each is 24 cm in a circle of radius 13 cm, find that distance.
Equal chords are equidistant from the centre. Half = 12, d = √(13² − 12²) = √(169 − 144) = √25 = 5 cm.

✏️ Practice — try these, take hints as needed

1. A chord of length 24 cm is 5 cm from the centre. Find the radius.
Half-chord = 12.
r² = d² + half².
r = √(25 + 144).
13 cm
2. In a circle of radius 25 cm, a chord is 7 cm from the centre. Find the chord length.
Half = √(r² − d²).
√(625 − 49).
= 24, double it.
48 cm
3. Two parallel chords of lengths 16 cm and 12 cm lie on the same side of the centre of a circle of radius 10 cm. Find the distance between them.
Distances: √(100−64)=6 and √(100−36)=8.
Same side ⇒ subtract.
8 − 6.
2 cm
4. A chord subtends a right angle at the centre of a circle of radius 8 cm. Find the chord length.
Central angle 90°.
Chord = side opposite, use the isosceles right triangle.
chord = r√2.
8√2 cm
5. The longest chord of a circle is 26 cm. A chord 5 cm from the centre has what length?
Longest chord = diameter ⇒ r = 13.
Half = √(169 − 25).
√144 = 12, double.
24 cm

📝 Topic test — 8 questions

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