PT is a tangent drawn from an external point P, touching a circle at T. A secant through P meets the circle first at A and then at B. If PT = 12 cm and the chord AB measures 7 cm, what is the length of PA?

The tangent–secant relation gives $P{T}^2=PA\times PB$, and $PB=PA+AB=PA+7$. So $144=PA(PA+7)$, that is $P{A}^2+7PA-144=0$. Solving, $PA$ = (−7 + √(49 + 576))/$2=(-7+25)/2=9$ cm, and then $PB=16$ cm. Check: $9\times 16=144=12^2$. The trap 16 cm answers $PB$, the far intersection, rather than the near one; 12 cm copies the tangent length across; and 5 cm comes from the invalid subtraction 12 − 7.