P is a point inside a circle of radius 13 cm, and P is 5 cm from the centre. A chord drawn through P is cut by P into two pieces, one of which measures 9 cm. How long is the other piece?

For any chord through an interior point P, the product of the two pieces is the same number — the power of the point, $r^2-d^2$ where $d$ is the distance from the centre. Here that is $13^2-5^2=169-25=144$. So the second piece = 144/9 = 16 cm. A check: the chord is 25 cm long, comfortably under the diameter of 26 cm, its midpoint lies 3.5 cm from P, and √$(12.75+12.25)=5$ cm as required. The trap 12 cm takes √144 instead of dividing by 9, 18 cm just doubles the given piece, and 4 cm subtracts 9 from the radius.