Find the length of the chord of a circle where the radius is 7 cm and perpendicular distance is 6 cm.
Answer: Half-chord = √(72 − 62) = √13 cm, so the chord = 2√13 cm ≈ 7.21 cm.
Step-by-step solution
Given: Radius OB = 7 cm; Perpendicular distance OM = 6 cm
To find: Length of the chord AB
To find: Length of the chord AB
Idea: The perpendicular from the centre bisects the chord, making a right triangle with the radius as hypotenuse, the distance as one leg and half the chord as the other.
- Let AB be the chord and OM ⟂ AB. The perpendicular from the centre bisects the chord, so AB = 2MB.½ mark
- In right ΔOMB: MB2 = OB2 − OM2 = 72 − 62 = 49 − 36 = 13, so MB = √13 cm.1 mark
- AB = 2√13 cm ≈ 2 × 3.606 = 7.21 cm.½ mark
The chord is 2√13 cm ≈ 7.21 cm long.
Check: (√13)2 + 62 = 13 + 36 = 49 = 72 ✓.
Answer to write in the exam
OB = 7 cm, OM = 6 cm, OM ⟂ AB
MB2 = OB2 − OM2 (Baudhāyana–Pythagoras theorem)
= 49 − 36 = 13 ⇒ MB = √13 cm
AB = 2MB (perpendicular from the centre bisects the chord)
∴ AB = 2√13 cm ≈ 7.21 cm
Common mistakes that cost marks
- Giving √13 cm as the answer. That is only half the chord; double it.
- Adding instead of subtracting: √(49 + 36). The radius is the hypotenuse, so subtract.
- Rounding √13 to 3 or 4 too early; keep 2√13 or use 3.606.
How this can come in the exam
MCQ (1 mark)
A chord of a circle of radius 10 cm is 8 cm from the centre. Its length is
- 6 cm
- 12 cm
- 18 cm
- 2√41 cm
Show answer
(B) 12 cm
Half-chord = √(100 − 64) = 6 cm, so the chord is 12 cm.
Try one yourself
Find the length of a chord of a circle of radius 9 cm that is 5 cm from the centre.
Show answer
Half-chord = √(81 − 25) = √56 = 2√14 cm, so the chord = 4√14 cm ≈ 14.97 cm.
More questions like this
- 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.
- Draw a circle and a chord AB. Fix an arc AKB formed by AB and a point K between A, B on the circle. Measure the angle subtended at the centre by arc AKB. Take three points P, Q, R on the circle outside arc AKB. Measure the angles subtended by arc AKB at points P, Q, R. What do you notice?
Repeat this activity for a different arc AKB. - The angle subtended by an arc at the centre of the circle is double the angle subtended by the arc at any point on the circle outside the arc.