+223 Two tangents \(\overline{PA}\) and \(\overline{PB}\) are drawn to a circle, where $P$ lies outside the circle, and $A$ and $B$ lie on the circle. The length of $PA$ is $12,$ and the circle has a radius of $9.$ Find the length $AB.$
+23245 PA and PB are tangents to circle(O). Draw PO. Draw AB. Let X be the point where PO and aB intersect.
PA = 12 and OA = 9
Because triangle(OPA) is a right triangle, PO = 15.
Triangle(AXO is similar to triangle(PAO) ---> AX / AO = PA / PO ---> AX / 9 = 12 / 15
---> AX = 7.2
Since X is the midpoint of AB ---> AB = 14.4.
