Cyclic quadrilaterals - Higher - Circle theorems - Higher - OCR - GCSE Maths Revision - OCR

Cyclic quadrilaterals - Higher

Find the refreshed revision resources for GCSE Maths: Cyclic quadrilaterals, with step-by-step slideshows, quizzes, practice exam questions, and more!

A cyclic quadrilateral is a drawn inside a circle. Every corner of the quadrilateral must touch the of the circle.

The second shape is not a cyclic quadrilateral. One corner does not touch the circumference.

The opposite angles in a cyclic quadrilateral add up to 180°.

\(a + c = 180^\circ\)

\(b + d = 180^\circ\)

Cyclic quadrilateral with angles a, b, c and d

Example

Calculate the angles \(a\) and \(b\).

Cyclic quadrilateral with angles a, b, 60 degrees and 140degrees

The opposite angles in a cyclic quadrilateral add up to 180°.

\(b = 180 - 140 = 40^\circ\)

\(a = 180 - 60 = 120^\circ\)

Proof

Let angle CDE = \(x\) and angle EFC = \(y\).

Cyclic quadrilateral (angles x and y at the circumference)

The angle at the centre is double the angle at the circumference.

Angle COE = \(2y\) and the reflex angle COE = \(2x\).

Cyclic quadrilateral with angles x and y at the circumference and 2x and 2y at the centre

Angles around a point add up to 360°.

\(2y + 2x = 360^\circ\)

\(\frac{2y}{2} + \frac{2x}{2} = \frac{360^\circ}{2}\)

So \(y + x = 180^\circ\)

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Circle theorems - Higher - OCRCyclic quadrilaterals - Higher