Central angles, arcs and chords

An arc is measured in degrees, by the central angle standing on it: the measure of an arc is its central angle.

• In the drawing $\displaystyle {\angle AOB}={70^{\circ}}$, so the minor arc $\displaystyle AB$ is $\displaystyle 70^{\circ}$.
• The chord $\displaystyle AB$ cuts the circle into two arcs: the minor arc and the major arc.
• The whole circle is $\displaystyle 360^{\circ}$, so the major arc $\displaystyle AB$ is $\displaystyle {360^{\circ}-70^{\circ}}={290^{\circ}}$.

Theorem: In a circle, two central angles are equal if and only if their chords are equal.

“If and only if” is two theorems: from equal angles we learn that the chords are equal, and from equal chords that the angles are equal. Here we prove the first.

Given: $\displaystyle O$ is the centre of the circle, $\displaystyle {\angle AOB}={\angle COD}$.
To prove: $\displaystyle {AB}={CD}$.

The proof below shows that the triangles $\displaystyle AOB$ and $\displaystyle COD$ are congruent: all their arms are radii, and the angles between them are equal.

• The other direction – from equal chords to equal central angles – you prove yourself, in Exercise 3.
• In a circle, if two arcs are equal, then their chords are equal. Equal arcs are arcs with equal central angles, so this follows from the theorem we proved.

Theorem: The perpendicular to a chord through the centre of the circle bisects the chord’s central angle and its arc.

Given: $\displaystyle O$ is the centre of the circle, $\displaystyle OM\perp AB$, and $\displaystyle M$ is on the chord $\displaystyle AB$.
To prove: $\displaystyle {\angle AOM}={\angle BOM}$.

The proof below shows that the triangles $\displaystyle OMA$ and $\displaystyle OMB$ are congruent. The central angles $\displaystyle \angle AOM$ and $\displaystyle \angle BOM$ are equal, so their arcs are equal too: $\displaystyle OM$ extended meets the arc $\displaystyle AB$ at its middle.

$\displaystyle O$ is the centre of the circle, and $\displaystyle {\angle AOB}={110^{\circ}}$.

The points $\displaystyle A$, $\displaystyle B$, $\displaystyle C$ cut the circle into three arcs.
The measures of the arcs $\displaystyle AB$, $\displaystyle BC$, $\displaystyle CA$ are in the ratio $\displaystyle 2:3:4$.
$\displaystyle O$ is the centre of the circle.

$\displaystyle O$ is the centre of the circle, and the chords $\displaystyle AB$ and $\displaystyle CD$ are equal.
Prove: $\displaystyle {\angle AOB}={\angle COD}$.

$\displaystyle O$ is the centre of the circle, and the chords $\displaystyle AB$ and $\displaystyle CD$ are equal.
$\displaystyle OM\perp AB$ and $\displaystyle ON\perp CD$; $\displaystyle M$ is on $\displaystyle AB$, and $\displaystyle N$ on $\displaystyle CD$.
Prove: $\displaystyle {OM}={ON}$ – equal chords are equally far from the centre.

A circle of radius $\displaystyle 5$ cm has a chord $\displaystyle AB$ of length $\displaystyle 8$ cm.
$\displaystyle O$ is the centre of the circle, $\displaystyle OM\perp AB$, and $\displaystyle M$ is on $\displaystyle AB$.