SOLUTION: The diagonals of rhombus ABCD meet at M. Angle DAB measures 60 degrees. Let P be the midpoint of AD and let G be the intersection of PC and MD. Given that AP = 8, find MD, MC, MG,

Question 148142: The diagonals of rhombus ABCD meet at M. Angle DAB measures 60 degrees. Let P be the midpoint of AD and let G be the intersection of PC and MD. Given that AP = 8, find MD, MC, MG, CG, and GP.
thank you all veryyyy much !
Answer by orca(409) (Show Source): You can put this solution on YOUR website!
The two diagonals are perpendicular and bisect each other.
As AB = AC and < A = 60, Triangle ABD is equilateral.
AB = BD = AD = 2*AP = 16
MD = BD/2 = 8
MC = AM = sqrt(AD^2 - MD^2) = sqrt(16^2 - 8^2) = 8*sqrt(3)
In triangle ADC, MD and PC are medians, G is centroid of the triangle. So
MG = MD/3 = 8/3
CG = sqrt(MC^2 + MG^2)
= sqrt{[8sqrt(3)]^2 + (8/3)^2}
= sqrt[(8^2)(3 + 1/9]
= sqrt[(8^2)(28/9]
= (8/3)sqrt(28)
= (16/3)sqrt(7)
As G is the centroid, GP = CG/2 = (8/3)sqrt(7)

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