Circles

Tangents to a circle: a tangent is at right angles to the radius, and the two tangents from an outside point are equal.

Key ideas

A tangent meets a circle at exactly one point, the point of contact; a secant meets it at two points.
From a point inside a circle there is no tangent, from a point on it one, and from a point outside it two.
The tangent at any point is perpendicular to the radius through the point of contact.
The two tangents from an outside point are equal in length.
The line from the outside point to the centre bisects the angle between the two tangents.
A quadrilateral drawn around a circle has AB + CD = AD + BC.
Most numerical questions are Pythagoras in the right triangle made by the radius, the tangent and the line to the centre.

Tangent length	PT = √(OP² − r²)
Angle between tangents	∠APB + ∠AOB = 180°
Circumscribed quadrilateral	AB + CD = AD + BC

Using OP² = PT² − r² (the hypotenuse is OP, the line to the centre).
Thinking the tangent passes through the centre.
Forgetting that both tangents from the same point are equal when splitting the sides of a triangle.

From a point P, 50 cm from the centre O of a circle of radius 40 cm, tangents PA and PB are drawn. Find the length of AB.

Working: PA = 30; OP bisects AB at right angles, and (1/2) × OA × PA = (1/2) × OP × (AB/2), so AB = 2 × 40 × 30/50

Answer: 48 cm