SOLUTION: If a 30° - 60° - 90° triangle has a perimeter of 12 units and its area is in the form of (a√b - c), then determine the numerical value of (a + b + c).

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Question 256013: If a 30° - 60° - 90° triangle has a perimeter of 12 units and its area is in the form of (a√b - c), then determine the numerical value of (a + b + c).
Answer by drk(1908) About Me  (Show Source):
You can put this solution on YOUR website!
Let side opposite 90 = x
Let side opposite 30 = (1/2)x
Let side opposite 60 = (sqrt(3)/2)x
Perimeter = 12, so
(i) x+%2B+%281%2F2%29x+%2B+%28sqrt%283%29%2F2%29x+=+12
solving for x, we get
(ii) %283%2F2%29x+%2B+%28sqrt%283%29%2F2%29x+=+12
multiply by 2 to get
(iii) 3x+%2B+x%2Asqrt%283%29+=+24
factor out an x to get
(iv) x%283%2Bsqrt%283%29%29+=+24
dividing, we get
(v) x+=+24%2F%283%2Bsqrt%283%29%29
So,
side opposite 90 = 24/(3+sqrt(3))
side opposite 30 = 12/(3+sqrt(3))
side opposite 60 = 12sqrt(3)/(3+sqrt(3))
using these 3 lengths, our perimeter =12.
we want a + b + c values. I think we have them. The area formula is a bit confusing.