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# Circle tangency

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Two distinct points $A$ and $B$ are on a circle with center at $O$, and point $P$ is outside the circle such that $\overline{PA}$ and $\overline{PB}$ are tangent to the circle. Find $AB$ if $PA = 12$ and the radius of the circle is 9.

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#1
+76198
+1

OA is a radius = 9

Because  OAP  is a right angle, OP  = sqrt ( PA^2 + OA^2)  = sqrt (12^2 + 9^2)  =

sqrt (225)  =  15

Let AB intersect OP at R

And triangles  OPA and OAR  are similar

So

PA / OP  =  AR / OA

12 / 15  =  AR / 9

AR  = 108/15  =  36/5

And AR  = (1/2) AB  ...so

AB  = 2 (AR) =  2(36/5)   =  72/5

CPhill  Sep 3, 2017
#2
+18566
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Two distinct points $A$ and $B$ are on a circle with center at $O$, and point $P$ is outside the circle such that $\overline{PA}$ and $\overline{PB}$ are tangent to the circle. Find $AB$ if $PA = 12$ and the radius of the circle is 9.

heureka  Sep 4, 2017

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