From a point P outside a circle with center O, tangents PA and PB are drawn to the circle. The line segment formed by joining the points A and B intersect the line segment OP at M. Find the measure of AMP.
A O P B M


Answer:

90

Step by Step Explanation:
  1. Given:
    PA and PB are the tangents to the circle from an external point P.

    To find:
    The measure of AMP.
  2. In MAP and MBP, we have PA=PB[ Tangents to a circle from an external point are equal.]MP=MP[Common side]OPA=OPB[Tangents from an external point are equally inclined to    the line segment joining the center to that point.]MPA=MPB[As, OPA=MPA and OPB=MPB] Thus, MAPMBP   [By SAS-congruence]
  3. As the corresponding parts of congruent triangles are equal, MA=MB and AMP=BMP.

    Also,  AMP+BMP=180[Angles on a straight line.]AMP+AMP=180[As, AMP=BMP]2AMP=180AMP=1802=90AMP=BMP=90
  4. Therefore, the measure of AMP is 90.

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