8.03 Chords and arcs

The diagram shows two equal chords on a circle, joined to the centre by some radii.

Since radii in a circle are equal, we know that OA, \, OB, \, OC and OD are all equal. We can write this information as:\text{ } \\ \begin{array}{cll} OA = OC &\text{(Radii of a circle are equal)} \\ OB = OD &\text{(Radii of a circle are equal)} \\ \end{array}

Since we are given that the chords are equal, we also know that:\text{ } \\ \begin{array}{cll} AB = CD &\text{(Given)} \\ \end{array}

So \triangle OAB \equiv \triangle OCD by SSS.

In our four triangles, we know that:

• AO, \, BO, CO and DO are all equal radii.

• \angle OAX, \, \angle OBX, \, \angle OCY and \angle ODY are all equal since they are base angles in congruent isosceles triangles.

• \angle OXA, \, \angle OXB, \, \angle OYC and \angle OYD are all right angles.

So all four triangles are congruent under the test AAS.

In the diagram, we have an angle at the centre and an angle at the circumference subtended by the same chord.

Since they are all radii, we know that OA, \, OB and OC are all equal and that \triangle OAC and \triangle OBC are both isosceles.

Given that \angle OCA = x, then \angle OAC = x (base angles in isosceles triangles are equal).

Similarly, given that \angle OCB = y, then \angle OBC = y.

It is worth noting that if we move the point C too far around the circle, the point P will no longer lie on the arc AB. In order to prove our angle relation for this case, we need to construct an additional line as shown.

If we let \angle OCB=x and \angle ACB=y, we can use a similar proof except that it will involve subtraction instead of addition.

Explore the above proof using the applet. Move the points on the circle.

The angle at the centre of a circle is always twice the angle at its circumference that stands on the same arc.

Properties of equal chords:

• Chords of equal length subtend equal angles at the centre.

• Chords of equal length are equidistant from the centre.

• The perpendicular line from a centre of a circle to a chord bisects its chord.

• The line from the centre of a circle to the midpoint of a chord is perpendicular to the chord.

• The perpendicular bisector of a chord must pass through the centre of the circle.

• Angle at the centre of a circle is twice the angle at its circumference.

Consider these angles subtended by the same arc.

Consider that angle \angle AOB is the angle at the centre with respect to \angle ADB, we can see that \angle AOB=2 \angle ADB.

Also, \angle AOB is the angle at the centre with respect to \angle ACB, so \angle AOB=2 \angle ACB.

Since \angle ADB and \angle ACB are both equal to half \angle AOB, they must be equal.

In this case, we can see that the angle at the centre is a straight angle. In other words, the angle at the centre is 180\degree.

Since the angle at the circumference must be equal to half of this, we find that any angle standing on the diameter must be a right angle.

Angle properties:

• Angles standing on equal length chords or arcs are equal.

• Angles in the same segment are equal.

• The angle in a semicircle is a right angle.