You have two chords on a circle. One is longer than the other. Which chord is closer to the centre? Can you guess?
Activity: Draw a circle. Draw chords of various lengths. Drop a perpendicular to each chord from the centre. Record the length of the chord and its distance from the centre in a table.
What do you observe?
Step-by-step solution
Idea: For a chord of length l at distance d from the centre of a circle of radius r, (l/2)2 + d2 = r2. With r fixed, a bigger l leaves less room for d.
- Guess: the longer chord is closer to the centre (the longest chord, the diameter, passes through the centre).½ mark
- Drawing: draw a circle of radius 5 cm with centre O. Draw chords of different lengths, and from O drop a perpendicular to each with a set square. Measure each chord and each perpendicular.½ mark
- A sample table (radius 5 cm):
Length of chord: 10 cm | 9.6 cm | 8 cm | 6 cm | 2.8 cm
Distance from centre: 0 cm | 1.4 cm | 3 cm | 4 cm | 4.8 cm1 mark - Observation: as the chord gets longer, its distance from the centre gets smaller. The longer the chord, the closer it is to the centre. The diameter (10 cm) is at distance 0.½ mark
- Check one row: chord 8 cm, half = 4 cm: 42 + 32 = 25 = 52 ✓. So each row fits (half-chord)2 + (distance)2 = (radius)2.½ mark
Check: Every row satisfies (½ chord)2 + distance2 = 25: 4.82 + 1.42 = 23.04 + 1.96 = 25 ✓; 1.42 + 4.82 = 25 ✓; 32 + 42 = 25 ✓.
Answer to write in the exam
Radius = 5 cm.
Chord 10 cm → distance 0 cm
Chord 9.6 cm → distance 1.4 cm
Chord 8 cm → distance 3 cm
Chord 6 cm → distance 4 cm
Chord 2.8 cm → distance 4.8 cm
Check: 42 + 32 = 52
∴ The longer the chord, the closer it is to the centre.
Common mistakes that cost marks
- Measuring the distance from the centre to an end of the chord (always the radius) instead of along the perpendicular.
- Drawing the perpendicular by eye. Use a set square; a slanting line is longer than the true distance.
- Concluding the opposite (longer chord farther away) from badly drawn chords. Check one row with (½ chord)2 + distance2 = radius2.
How this can come in the exam
Four chords of one circle have lengths 7 cm, 9 cm, 4 cm and 8 cm. Which chord is nearest to the centre?
- 7 cm
- 9 cm
- 4 cm
- 8 cm
Show answer
(B) 9 cm
The longer the chord, the nearer it is to the centre, so the 9 cm chord is nearest.
Try one yourself
In a circle of radius 53 cm, find the distances from the centre of chords of length 90 cm and 56 cm. Which is nearer the centre?
Show answer
90 cm chord: √(532 − 452) = √784 = 28 cm. 56 cm chord: √(532 − 282) = √2025 = 45 cm. The longer (90 cm) chord is nearer.
More questions like this
- Let AB and DE be two chords of a circle with centre C. Suppose AB > DE. Then the distance from C to AB is less than the distance from C to DE.
- Find the length of the chord of a circle where the radius is 7 cm and perpendicular distance is 6 cm.
- Explain why the following statement is true: If the perpendicular distance of a chord from the centre is d and the radius is r, then the chord length is 2√(r2 − d2).
- In a circle, if the distance of chord AB from the centre is twice the distance of another chord CD from the centre, then can we conclude that CD = 2 AB? Give reasons for your answer.
- A circle with centre O is drawn, and A, B, C, D are points on the circle (see the figure). Measure the angles subtended by arc AKB and arc CLD at the centre O. If the angle at the centre is less than 180°, it is a minor arc. If the angle at the centre is greater than 180°, it is a major arc. State whether arcs AKB and CLD are minor arcs or major arcs.