Chords of a circle having the same length are all at the same distance from the centre of the circle.

Step-by-step solution

Idea: The distance of a chord from the centre is the perpendicular from the centre, and that perpendicular meets the chord at its midpoint. Two equal chords give two congruent right triangles (half-chord, distance, radius), so the distances match. Two ways of showing this are given below.

1. E and H are midpoints, so CE ⟂ AB and CH ⟂ FG (the line from the centre to the midpoint of a chord is perpendicular to it). So CE and CH are the distances of the chords from C.1 mark
2. Way 1 (congruent triangles): CA = CF and CB = CG (radii) and AB = FG (given), so ΔCAB ≅ ΔCFG (SSS). Congruent triangles have equal altitudes on matching sides, so CE = CH.
3. Way 2 (right triangles): In ΔCEA and ΔCHF: AE = ½AB = ½FG = FH (E, H are midpoints and AB = FG).½ mark
4. ∠CEA = ∠CHF = 90°, and hypotenuse CA = hypotenuse CF (radii).1 mark
5. So ΔCEA ≅ ΔCHF (RHS congruence rule).1 mark
6. Hence CE = CH (CPCT). The chords are equidistant from the centre.½ mark

Answer to write in the exam

Common mistakes that cost marks

• Measuring the distance of a chord to one of its end points instead of along the perpendicular.
• Using SAS with the right angles but taking CE = CH as a side; that is what must be proved.
• Forgetting to justify AE = FH: it needs both AB = FG and the fact that E and H are midpoints.

How this can come in the exam

Try one yourself

In a circle with centre O, chords PQ and RS are equal and ∠POQ = 80°. OX ⟂ PQ and OY ⟂ RS. Compare OX and OY and find ∠ROS.

More questions like this

• Use the Baudhāyana–Pythagoras theorem to show why chords of a circle having the same length are all at the same distance from the centre of the circle.
• Consider the figure. If CE is perpendicular to AB, CH is perpendicular to GH, and CE = CH, show that AB = GF.
• If CE is perpendicular to AB, CH is perpendicular to GH, and CE = CH, show that AB = GF. Solve this using the Baudhāyana–Pythagoras theorem.
• 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?
• 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.

All Circles questions · All maths questions