This problem becomes easier if we lay it out like this :
Let D = (0,0)
A = (-24 cos60, 24 sin 60) = (-12, 12√3)
C = (24 cos 60, 24 sin 60) = (12, 12√3)
Let B = (0, y)
To find y....we can use the square of the distance formula
(12 - 0)^2 + (y - 12√3)^2 = 13^2
12^2 + (y - 12√3)^2 = 13^2
(y - 12√3)^2 = 13^2 - 12^2
(y - 12√3)^2 = 25 take the square root of both sides
y - 12√3 = 5
y = = 5 + 12√3
So....B = (0, 5 + 12√3)
X = [ (12 + 0)/2, (12√3 + 5 + 12√3)/2 ] = (6, 2.5 +12√3)
Y = (-6, 6√3)
So XY^2 = (6 - - 6)^2 + ( 2.5 + 12√3 - 6√3)^2 =
(12)^2 + (2.5 + 6√3)^2 =
144 + 6.25 + 30√3 + 108 =
258.25 + 30√3 units ≈ 310.212 units