“There is no chord of a circle that is longer than its diameter.” How do you justify this statement?
Answer: For any chord AB of a circle with centre O and radius r, the triangle inequality gives AB ≤ OA + OB = 2r, with equality only when O lies on AB, that is, when AB is a diameter. So no chord is longer than a diameter.
Step-by-step solution
Given: A circle with centre O and radius r; Any chord AB
To find: Show that AB ≤ 2r
To find: Show that AB ≤ 2r
Idea: Compare the chord with the path from A to B through the centre. Or use the distance formula: a chord at distance d from the centre has length 2√(r2 − d2), which is largest when d = 0.
- If O is not on AB, then in ΔOAB the sum of two sides is greater than the third: AB < OA + OB = r + r = 2r.1 mark
- If O is on AB, then AB = OA + OB = 2r: AB is a diameter. So every chord satisfies AB ≤ 2r = diameter: no chord is longer than the diameter.1 mark
- Another way: a chord at distance d from O has length 2√(r2 − d2) ≤ 2√(r2) = 2r, with equality only for d = 0.
Any chord AB satisfies AB ≤ OA + OB = 2r (triangle inequality), with equality only when the chord passes through the centre. So the diameter is the longest chord.
Answer to write in the exam
Let AB be a chord of a circle with centre O and radius r.
If O is not on AB: AB < OA + OB = 2r (triangle inequality in ΔOAB)
If O is on AB: AB = OA + OB = 2r (AB is a diameter)
∴ AB ≤ 2r = diameter; no chord is longer than a diameter.
Common mistakes that cost marks
- Saying “the diameter goes through the middle so it is longest” without a reason. Use the triangle inequality or the formula 2√(r2 − d2).
- Forgetting the equality case: a chord through the centre equals the diameter (it is a diameter).
- Writing AB = OA + OB for every chord; that holds only when O is on AB.
How this can come in the exam
MCQ (1 mark)
Which of these cannot be the length of a chord of a circle of radius 7 cm?
- 5 cm
- 10 cm
- 14 cm
- 15 cm
Show answer
(D) 15 cm
The longest chord is the diameter, 14 cm, so a 15 cm chord is impossible.
Try one yourself
Can a circle of radius 4.5 cm have a chord of length 9.5 cm? Explain.
Show answer
No. The longest chord is the diameter, 9 cm, and 9.5 cm is longer than that.
More questions like this
- Let A be any point within a given circle with centre O. Show that the shortest chord of the circle that passes through point A is the one that is perpendicular to OA.
- How would you use the following figure to justify the statement that the angle in a semicircle is 90°?
- In a circle, two chords CC’ and DD’ are drawn perpendicular to a diameter AB. Prove that the segment MM’ joining the midpoints of the chords CD and C’ D’ is perpendicular to AB.
- How would you use the following figure to justify the statement that the sum of the opposite angles of a cyclic quadrilateral is 180°?
- Can you recognise the origin of the shapes in the figure?