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Chords $\overline{AB}$ and $\overline{CD}$ of a circle meet at $Q$ inside the circle. If $CD = 38$, $AQ = 6$, and $BQ = 12$, then what is the minimum length of $CQ$?

 Feb 13, 2020
 #1
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Intersecting chord theorem :

 

AQ * BQ   =  CQ * DQ

 

Let  CQ  = x

Then since CD = 38, then  DQ  = (38 - x)

 

So......

 

6 * 12  =   x (38 - x)

 

72  = 38x - x^2       rearrange  as

 

x^2 - 38x  + 72   = 0     factor

 

(x  - 36)  ( x - 2)  = 0

 

Set each factor to  0    and solve for  x and we get that

 

x = 36    or  x  = 2

 

Then  the  minimum length of   CQ  =   2

 

 

cool cool cool

 
 Feb 13, 2020

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