Circle theorems I Revision Notes

When a triangle is formed by a diameter of a circle and any point on the circle, the angle opposite the diameter is always a right angle (90°). This is because the diameter subtends a semicircle, and the inscribed angle on a semicircle is a right angle.

To use this theorem, identify the diameter and the point on the circle forming the triangle. The angle at that point is 90°, which helps find unknown angles or confirm right angles in problems.

• If AB is a diameter and C lies on the circle, then ∠ACB = 90°.
• This theorem is useful for proving triangles are right-angled in circle problems.

A tangent touches a circle at exactly one point. The radius drawn to the point of contact is perpendicular to the tangent line, forming a 90° angle.

This property allows you to find unknown angles involving tangents by recognising right angles where the radius meets the tangent.

For example, if a tangent touches the circle at point T and the centre is O, then ∠OTTangent = 90°. This can be used to calculate adjacent angles in the diagram.

• Tangent at point T and radius OT satisfy ∠OTTangent = 90°.
• Use this to find unknown angles adjacent to the tangent or radius.

An arc of a circle subtends two angles: one at the centre and one at the circumference. The angle at the centre is always twice the angle at the circumference subtended by the same arc.

This relationship helps find unknown angles when both centre and circumference angles are involved.

For example, if arc AB subtends ∠AOB at the centre and ∠ACB at the circumference, then ∠AOB = 2 × ∠ACB.

• If ∠ACB = 30°, then ∠AOB = 60°.
• Use this to relate central and inscribed angles for calculations.

Angles subtended by the same chord and on the same side of the chord are equal. This means if two points lie on the circumference on the same side of a chord, the angles they form with the chord are equal.

This property is useful for finding unknown angles or proving angle equality in circle problems.

For example, if chord AB subtends angles ∠ACB and ∠ADB on the same segment, then ∠ACB = ∠ADB.

• Angles subtended by the same chord in the same segment are equal.
• Use this to find missing angles or prove angle relationships.

A cyclic quadrilateral is a four-sided figure with all vertices on the circle. The opposite angles of such a quadrilateral add up to 180° (are supplementary).

This property helps find unknown angles inside cyclic quadrilaterals and is often used to justify angle sums.

For example, in cyclic quadrilateral ABCD, ∠A + ∠C = 180° and ∠B + ∠D = 180°.

• Opposite angles in cyclic quadrilaterals sum to 180°.
• Use this to calculate unknown angles or verify cyclic quadrilaterals.

This theorem links the angle between a tangent and a chord at the point of contact to the angle in the alternate segment of the circle.

Specifically, the angle between the tangent and chord equals the angle subtended by the chord in the opposite segment of the circle.

For example, if a tangent touches the circle at point A and chord AB is drawn, then the angle between the tangent and chord at A equals the angle subtended by chord AB in the alternate segment (not containing the tangent).

• Angle between tangent and chord = angle in alternate segment.
• Use this to find unknown angles involving tangents and chords.