Find \(AC\).
https://latex.artofproblemsolving.com/9/1/d/91d520533f0fb8cb325932013b2aebf0db9bde54.png
+128475 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
