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# Circles

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Let line AB and line CD be chords of a circle, that meet at the point Q inside the circle. If AQ = 16, BQ = 12, and CD = 36, then find the minimum length of CQ.

Jun 17, 2021

#1
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Intersecting Chord Theorem

AQ * BQ  =  CQ  *  DQ

16  *  12   =  CQ  ( 36   -  CQ)

192   =  36CQ   -  CQ^2        rearrange  as

CQ^2   -  36CQ    +  192     =   0         complete  the  square on CQ

CQ^2   - 36CQ    =  -192

CQ^2   -  36CQ   +  324    =   -192  +  324

(CQ   -  18)^2   =   132    take  the negative root

CQ -  18  =  -sqrt (132)

CQ  -  18  =  -2sqrt 33

CQ =  18  -  2sqrt (33)  ≈     6.51 =    min length of CQ

EDIT   TO   CORRECT   A  PREVIOUS   ERROR    !!!!!   Jun 17, 2021
edited by CPhill  Jun 17, 2021
#3
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36 - 4.144 = 31.856

4.144 * 31.856 = 132.011264

civonamzuk  Jun 17, 2021
edited by civonamzuk  Jun 17, 2021
#2
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Let line AB and line CD be chords of a circle, that meet at the point Q inside the circle. If AQ = 16, BQ = 12, and CD = 36, then find the minimum length of CQ.

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I didn't see the word integer

AQ * BQ = CQ * DQ

16 * 12 = CQ(36 - CQ)

CQ = 2(9 - √33)

Jun 17, 2021
#4
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THX,  civonamzuk for  spotting  my  earlier error   !!!!!   CPhill  Jun 17, 2021
#5
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Not a problem. civonamzuk  Jun 17, 2021