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avatar+1347 

Point $Y$ is on a circle and point $P$ lies outside the circle such that $\overline{PY}$ is tangent to the circle.  Point $A$ is on the circle such that segment $\overline{PA}$ meets the circle again at point $B$.  If $PA = 12$ and $PY = 6$, then what is $AB$?

 Aug 19, 2023
 #1
avatar+128758 
+1

             P

                 6       

12    B         Y

               

 

Secant-Tangent Theorem

 

Let  AB = x   and PB = 12 - x

 

We have that

 

PY^2  = PB ( PA)

 

6^2 = (12-x) (12)

 

36  = 144 - 12x

 

12x =  144 - 36

 

12x  = 108

 

x = 108 / 12    =  9   = AB

 

 

cool cool cool     

 Aug 19, 2023

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