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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 $AB.$

Jun 9, 2020

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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.

Jun 9, 2020