Circle angle and tangent properties

A tangent meets a radius at a right angle

A tangent at T is perpendicular to the radius OT at the point of contact. Two tangents PA and PB drawn from the same external point P have equal lengths. OP bisects the angle APB: the two right triangles share hypotenuse OP and have equal radii OA and OB.

Equal chords lie the same perpendicular distance from the centre. The perpendicular bisector of a chord passes through the centre. State which circle and centre are involved: matching lengths in different circles need not imply these relationships without equal radii.

Match angles to the same arc

An angle at the centre is twice an angle at the circumference standing on the same arc, with the angle regions chosen consistently. If the circumference angle is 37°, the corresponding centre angle is 74°.

Angles in the same segment, standing on the same chord from the same side, are equal. Angles standing on the chord from opposite segments total 180°. Thus opposite angles of a cyclic quadrilateral are supplementary. Being a quadrilateral inside a circle is not enough: all four vertices must lie on the circle.

An angle subtended by a diameter at another circumference point is 90°, since the corresponding centre angle is 180°. The vertex must be on the circumference for this theorem to apply.

Check your understanding 1 Locate the vertices

AB is a diameter of a circle and C lies on its circumference, distinct from A and B. Angle CAB is 28°. Find ACB and ABC with reasons.

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ACB is 90° because it stands on diameter AB. In triangle ABC, ABC is 180° − 90° − 28° = 62°. If C were inside the circle, the diameter theorem would not establish the right angle.

Combine tangent and centre angles

Tangents PA and PB touch a circle at A and B. The smaller angle AOB at its centre is 124°, and P is outside the circle in the usual tangent intersection region. Find angle APB.