From an external point P a secant meets a circle first at C and then at D, with PC = 3 cm and CD = 9 cm. A second secant through P meets the circle first at A and then at B, with PA = 4 cm. What is the length of the chord AB?

First find PD = PC + CD = 3 + 9 = 12 cm, so the power of the point is PC × PD = $3\times 12=36$. For the second secant, PA × PB = 36 gives 4 × PB = 36 and PB = 9 cm. The chord asked for is AB = PB − PA = 9 − 4 = 5 cm. Answering 9 cm gives PB, the whole secant, rather than the chord inside the circle.