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Find \(AC\).

 

https://latex.artofproblemsolving.com/9/1/d/91d520533f0fb8cb325932013b2aebf0db9bde54.png

 May 28, 2019
 #1
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The answer is 2 root 3

 May 28, 2019
 #2
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+1

We can use the Law of Cosines

 

BC^2  = AC^2 + AB^2 - 2(AC)(AB)cos(BAC)

 

(7√3)^2  = AC^2 + 9^2 - 2(AC) (9) (-√3/2)

 

147  = AC^2 + 81 + 9√3AC  subtract 81 from both sides

 

66 = AC^2 + 9√3AC

 

AC^2 + 9√3AC - 66 = 0        let  AC = x

 

x^2 + 9√3x - 66 = 0

 

Using the quadratic formula

 

-9√3 + √[ (9√3)^2 + 4*66]            -9√3 + √[169 *3 ]        -9√3 + 13√3                4√3

____________________    =  _______________  =   ____________  =       ____  =

             2                                           2                                 2                             2

 

 2√3  =  x  = AC

 

cool cool cool

 May 28, 2019

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