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# Help geometry

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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 PA and PB are tangent to the circle. Find AB if PA = 15 and the radius of the circle is 9.

May 28, 2021

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PA  =  PB  because   tangents  drawn  from an  external point  to  a  circle  are equal

Triangle   PAO  is  right with  angle PAO  = 90

So   PO   is  the  hypotenuse,     leg PA   = 15     and  leg  OA  =  the  radius

So

PO  = sqrt  ( PA^2   +  OA^2)   =  sqrt  ( 15^2  +  9^2)   =  sqrt  306   =  3sqrt 34

The  area  of  this  triangle   =    (15 * 9)  / 2    =    135/2

And    PBOA   will form a kite  with an area of  twice  [ PAO ]   =  135

And  PO  and  AB   will   be  the  diagonals  of  this kite

And  the  area   of  this kite  =   product of diagonal lengths  / 2

So

135   =   3sqrt 34  * AB  /  2

270   =  3sqrt 34 * AB

AB =   270  /  (3sqrt 34 )  =   90 / sqrt 34    units   ≈   15.43 units   May 28, 2021
edited by CPhill  May 28, 2021