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Let AB and CD be chords of a circle, that meet at point Q inside the circle. If AQ = 6, BQ = 15, and CD = 38, then find the minimum length of CQ.

 Jun 21, 2022
 #1
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Intersecting Chord Theorem :

 

AQ  * BQ = CQ * DQ

 

6  * 15  = CQ * DQ

 

Let CQ = x    and DQ = 38- x....so we have

 

6 * 15 = x  (38 - x)

 

90 =  38x - x^2            rearrange as

 

x^2 - 38x  + 90  =  0     complete the square on x

 

x^2  - 38x    =   -90

 

x^2 - 38x + 361  =  -90 + 361

 

(x - 19)^2   = 271          take the negative root

 

x - 19 = -sqrt (271)

 

x =  -sqrt (271) + 19 ≈  2.54 =  minimum length of CQ

 

 

 

cool cool cool

 Jun 21, 2022

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