Circle Theorems Part 3 - Tangents

Definition A tangent is a line which just touches the circumference of the circle once. O

Tangent Theorem 1 The radius drawn to the point of contact makes a right angle. O

Tangent Theorem 2 From any external point only two tangents can be drawn and they are of equal length. O

Alternate Segment Theorem The angle between a tangent and a chord at the point of contact is equal to the angle subtended by the chord in the alternate segment. These two angles are equal These two angles are also equal

Proof C D Draw a diameter AD. x x Angle DAT = 90 Angle DAB = 90 - x B Angle DBA = 90 Angle ADB = x O Angle ACB = x 90-x x A T

Example C a a = 70 (AST) b = 180 – 50 – 70 = 60 B b c = b = 60 (AST) O 50 c 70 A T