Q20
Q20 - In the diagram, circle O has a radius of 10, circle P is internally tangent to O and has a radius of 4. is tangent to circle P at T and, if drawn, line intersects circle O at points A and B. Compute the product TA•TB. Answer: 60 Solution: Extend QP and QT so that they intersect circle O at V and S, respectively. By the Power of a Point Theorem, in the big circle, Notice that QTS and ATB are chords in the big circle passing through the same point T, So AT.TB = QT.TS So if we find QT and TS, we are done. QV = 20(diameter of O). QP = QV - 4 = 16. Triangle QPT is right triangle so QT^2 = QP^2 - PT^2 = 256 - 16. QT = sqrt(240) = 4.sqrt(15). Now, QSV = 90 deg cause QV is the diameter subtending the angle at S. So QVS and QPT are similar. QP/QV = QT/QS = 4.sqrt(15)/QS = 16/20 QS = 5.sqrt(15) TS = QS- QT = sqrt(5) QT.TS = 4.5 = 20.