APAP and BP are the two tangents at the extremities of chord AB of a circle. Prove that ∠MAP is equal to ∠MBP.
Answer:
- Given:
AB is a chord of the circle with center O.
Tangents at the extremities of the chord AB meet at an external point P.
Chord AB intersects the line segment OP at M. - Now, we have to find the measure of ∠MAP.
In △MAP and △MBP, we have PA=PB[Tangents from an external point on a circle are equal in length] MP=MP[Common]∠MPA=∠MPB [Tangents from an external point are equally inclined to the line segment joining the point to the center.] ⟹△MAP≅△MBP [by SAS Congruency Criterion] - We know that corresponding parts of congruent triangles are equal.
Thus, ∠MAP=∠MBP.