AMSCO'S Geometry. New York - Rye High School

562 Geometry of the CircleGiven PQ gtangent to circle O at Q and PR gtangent tocircle O at R.QPProve PQ > PRORProof Draw OQ, OR, and OP. Since OQ and OR areboth radii of the same circle, OQ > OR. SinceQP and RP are tangent to the circle at Q andR, OQP and ORP are both right angles, and OPQ and OPR are righttriangles. Then OP is the hypotenuse of both OPQ and OPR. Therefore,OPQ OPR by HL. Corresponding parts of congruent triangles are congruent,so PQ > PR.The following corollaries are also true.Corollary 13.11a If two tangents are drawn to a circle from an external point, then the linesegment from the center of the circle to the external point bisects the angleformed by the tangents.Given PQ g tangent to circle O at Q and PR gtangentQPto circle O at R.ProveStrategyPO hbisects RPQ.Use the proof of Theorem 13.11 to show thatangles OPQ and RPO are congruent.ORCorollary 13.11b If two tangents are drawn to a circle from an external point, then the linesegment from the center of the circle to the external point bisects the anglewhose vertex is the center of the circle and whose rays are the two radiidrawn to the points of tangency.Given PQ g tangent to circle O at Q and PR gQtangentto circle O atPR.ProveStrategyOPhbisects QOR.Use the proof of Theorem 13.11 toshow that angles QOP and ROP arecongruent.OR