Two circles intersect at points P and Q. Their common tangents AB and CD meet the line PQ, extended on both sides, at points X and Y respectively. If PQ = 8 cm and XY = 17 cm, find the length of AB (in cm).

Solution

Solution: Using the standard common-tangent relationship for this configuration, \(XY^2=AB^2+PQ^2\). Therefore, \(AB^2=17^2-8^2=289-64=225\), so \(AB=\sqrt{225}=15\) cm. Trick: Square the given lengths, subtract the square of PQ from the square of XY, and take the square root.