AMSCO'S Geometry. New York - Rye High School

570 Geometry of the CircleEXAMPLE 1A tangent and a secant are drawn to circle O frompoint P. The tangent intersects the circle at Q andthe secant at R and S. If mQRX : mRS X : mSQ X 5 3:5:7 ,find:Xa. mQRXb. mRSc. mSQXQOSRPd. mQRS e. mRQP f. mPSolution Let mQRX 3x, mRS 5x, and mSQX 7x.X3x 1 5x 1 7x 5 36015x 5 360x 5 24a. mQR X 5 3x b. mRS X 5 5x c.5 3(24)5 5(24)mSQX 5 7x 7(24)5 725 1205 168m/P 5 1 X2 (mSQ 2 mQR X )d. m/QRS 5 1 X2 mSQ e. m/RQP 512 mQRX f.5 12 (168) 5 1 2 (72)5 1 2 (168 2 72)5 845 365 48Answers a. 72° b. 120° c. 168° d. 84° e. 36° f. 48°Note:m/QRS 5 m/RQP 1 m/P5 36 1 485 84EXAMPLE 2Two tangent segments, RP and RQ, are drawn to circle O from an external pointR. If mR is 70, find the measure of the minor arc PQX and of the major arc PSQinto which the circle is divided.X