Cyclic Quadrilaterals

Grade 6

What is a cyclic quadrilateral?

A cyclic quadrilateral is a four-sided polygon where all four vertices lie on the circumference of the same circle. Not every quadrilateral can be cyclic — the vertices have to fit on a single circle.

The key property: opposite angles in a cyclic quadrilateral add up to 180°. This means if you know one angle, you can find the opposite one immediately.

The exterior angle rule

There is a bonus rule: the exterior angle of a cyclic quadrilateral equals the interior opposite angle. This follows directly from the 180° rule.

Draw the circle around the quadrilateral in your working. It makes the cyclic quadrilateral property much easier to see and helps you spot which angles are "opposite".

Worked example

EXAMPLE

ABCD is a cyclic quadrilateral. Angle A = 78° and angle B = 105°. Find angles C and D.

Opposite angles sum to 180°. Angle C = 180° − 78° = 102°. Angle D = 180° − 105° = 75°. (Angle C = 102°, angle D = 75°).

Don't assume it's cyclic

Only apply the opposite angles rule if the question states or shows all four vertices on a circle. A regular quadrilateral drawn inside a circle in a diagram is not automatically cyclic unless the vertices touch the circle.

Try this one

A chord \( PQ \) is drawn in a circle of centre \( O \) and radius \( 9 \) cm. The perpendicular distance from \( O \) to the chord \( PQ \) is \( 4 \) cm.

Work out the length of the chord \( PQ \). Give your answer to 1 decimal place. [4]

Nova's hint:

The perpendicular from the centre to a chord bisects the chord, creating a right-angled triangle.

Ask yourself: 'If I drop a perpendicular from the centre, what right-angled triangle appears, and which side does Pythagoras give me?' The radius is the hypotenuse.

Try to Solve it with Nova?