Geometry
Circle
Alternate Segment Theorem
The angle between the tangent and a chord from the point of contact is equal to the angle subtended by the same chord in the alternate segment. In the diagram given below:
\(PAB\) is the alternate segment
\(PT\) is the tangent
\(PA\)
is the chord
\(\angle APT\, = \,\angle PBA\)
\(\angle T'PB\, = \,\angle PAB\)
Proof of Alternate Segment Theorem
Let \(T'PT\) is the tangent at the point \(P\) and \(O\) is the centre of the circle. The shaded part is the minor segment and \(PAB\) is the major segment of the circle. We have to prove that \(\angle APT=\angle PBA\).
Join the points \(P\) and \(A\) to centre \(O\), then the \(\angle POA = 2\angle PBA = 2x\).
Also \(OP = OA\), hence \(\angle OPA = \angle OAP = 90-x\).
Since the \(\angle TPO = 90\) hence \(\angle APT = x\)