Arcs and Chords | Moosa Academy

A chord is a segment whose two endpoints lie on a circle. How long a chord is, and the arc it cuts off, are both tied to its distance from the centre.

Theorem The diameter is the longest chord

The closer a chord is to the centre, the longer it is. Its length depends on the radius r and its distance d from the centre.

With r the radius and d the distance to the chord:

L = 2\sqrt{r^{2} - d^{2}}

When $d = 0$ the chord passes through the centre — that is the diameter, $2r$ .

Theorem Equal chords, equal arcs

Two chords are equal in length exactly when they are the same distance from the centre — and equal chords cut off equal arcs.

Equal distances give equal chords, and equal chords give equal arcs:

d_1 = d_2 \iff AB = CD

Theorem The perpendicular from the centre bisects the chord

A line from the centre that meets a chord at a right angle cuts the chord — and the arc between its endpoints — into two equal halves.

So $AM = MB$ , and the arc from A to B is halved at the point where the perpendicular meets the circle.

Only the perpendicularity matters — the full diameter is not required.

Example Three worked cases

Which chord is longest?

Circle of radius 5, with chords at distances 3, 4 and 0 from the centre.

Smaller d means longer chord, so

d = 0

wins:

L = 2\sqrt{5^2 - 0} = 10

⟹ the diameter (10) is the longest.

Using the perpendicular

A chord of length 24 is met by a perpendicular from the centre.

It bisects the chord: each part

= 24 \div 2

⟹ each half is 12.

Equal chords

Chords AB and CD are the same distance from the centre O.

Equal distances ⟹ equal chords.

⟹ AB = CD.

Summary

A chord has both endpoints on the circle; the longest one is the diameter, $2r$ .
Chord length: $L = 2\sqrt{r^2 - d^2}$ — the larger d is, the shorter the chord.
Chords equidistant from the centre are equal, and cut off equal arcs.
A perpendicular from the centre bisects the chord and its arc.