From a point PPP outside a circle with center OO, tangents PAPA and PBPB are drawn to the circle. The line segment formed by joining the points AA and BB intersect the line segment OPOP at MM. Find the measure of ∠AMP∠AMP.
Answer:
90∘90∘
- Given:
PAPA and PBPB are the tangents to the circle from an external point PP.
To find:
The measure of ∠AMP∠AMP. - In △MAP△MAP and △MBP△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, △MAP≅△MBP [By SAS-congruence]
- 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]⟹2∠AMP=180∘⟹∠AMP=180∘2=90∘⟹∠AMP=∠BMP=90∘ - Therefore, the measure of ∠AMP is 90∘.