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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
avatar+121006 
+1

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    !!!!!

 

 

cool cool cool

 Jun 17, 2021
edited by CPhill  Jun 17, 2021
 #3
avatar+1452 
+1

Check your answer again. Why didn't you use the quadratic formula to solve this problem?!?

 

36 - 4.144 = 31.856

 

4.144 * 31.856 = 132.011264

civonamzuk  Jun 17, 2021
edited by civonamzuk  Jun 17, 2021
 #2
avatar+1452 
+1

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. 

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

I didn't see the word integer

 

AQ * BQ = CQ * DQ

 

16 * 12 = CQ(36 - CQ)

 

CQ = 2(9 - √33)

 Jun 17, 2021
 #4
avatar+121006 
0

THX,  civonamzuk for  spotting  my  earlier error   !!!!!

 

 

cool cool cool

CPhill  Jun 17, 2021
 #5
avatar+1452 
+1

Not a problem. smiley

civonamzuk  Jun 17, 2021

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