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.
Answer:
90∘
- Given:
PA and PB are the tangents to the circle from an external point P.
To find:
The measure of ∠AMP. - 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, △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∘.