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Let $A$, $B$, $C$, and $D$ be points on a circle such that $AB = 11$ and $CD = 19.$ Point $P$ is on segment $AB$ with $AP = 7$, and $Q$ is on segment $CD$ with $CQ = 8$. The line through $P$ and $Q$ intersects the circle at $X$ and $Y$. If $PQ = 25$, find $XY$.

 Sep 6, 2023
 #1
avatar+128758 
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Let XQ =  x     and let PY  =  y

 

By the interesecting chord theorem we have two equations

 

PQ * QC =  XQ * QY

and

AP * PB  = PY * XP

 

So....subbing we have

11*8 = x * (25 + y)

7*4  =  y * (25 + x)             simplify these

 

88  = 25x + xy        (1)

28  = 25y  + xy        (2)

 

Subtract  (2)  from (1)

 

60  = 25 ( x - y)

60 / 25 = x - y

12/5  = x - y

2.4 + y  = x

 

28 = y * ( 25 + 2.4 + y)

28  = y * ( 27.4 + y)

28  = 27.4y + y^2

y^2 + 27.4y - 28 = 0

 

Solving this for positive y we have that y ≈ .986

And x = 2.4 + .986 ≈  3.386

 

XY  ≈ .986 + 25 + 3.386   ≈  29.372

 

cool cool cool

 Sep 7, 2023

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