Circles have different angle properties described by different circle theorems. Circle theorems are used in geometric proofs and to calculate angles.

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The perpendicular from the centre of a circle to a chord bisects the chord.

A circle has a radius of 5 cm. The chord EF is 7 cm.

How far is the midpoint of the chord from the centre of the circle?

Add the radii, OE and OF, to make two right-angled triangles.

FM is half of the length of chord EF.

FM = 3.5 cm

Use Pythagoras' theorem to calculate the length OM.

OF^{2} = FM^{2} + OM^{2}

OM^{2} =

OM^{2} = 12.75

OM = 3.6 cm (to 1 decimal place)

In the diagram below, AB is the chord of a circle with centre O.

OM is the perpendicular from the centre to the chord.

Angles OMA and OMB are both right-angles.

OA is the hypotenuse of triangle OAM.

OB is the hypotenuse of triangle OBM.

OA = OB as both are radii of the circle.

OM is common to both triangles.

Therefore, triangles OAM and OMB are congruent (RHS – right-angle, hypotenuse, side).

Therefore, the remaining sides of the triangles are equal, AM = MB.

So, M must be the mid-point of AB, and the chord has been bisected.