Question 1207270: Find the perimeter of an equilateral triangle whose area is equal to that of a triangle with sides 21 cm, 16 cm and 13 cm.
Answer by math_tutor2020(3817)   (Show Source):
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Triangle A = equilateral triangle
Triangle B = triangle with sides 21, 16, and 13.
Let's apply Heron's Formula to find the area of triangle B.
(a,b,c) = (21,16,13)
s = semi-perimeter = half the perimeter
s = (a+b+c)/2
s = (21+16+13)/2
s = 25
area = sqrt(s*(s-a)*(s-b)*(s-c))
area = sqrt(25*(25-21)*(25-16)*(25-13))
area = sqrt(25*4*9*12)
area = sqrt(10800)
area = 103.9230485 square cm approximately
Now we use the area of an equilateral triangle formula so we can find the side length of triangle A.
x = side length
area = 0.25*sqrt(3)*x^2
sqrt(10800) = 0.25*sqrt(3)*x^2
x^2 = 4*sqrt(10800)/sqrt(3)
x^2 = 240
x = sqrt(240)
x = sqrt(16*15)
x = sqrt(16)*sqrt(15)
x = 4*sqrt(15)
Each side of triangle A is exactly 4*sqrt(15) cm long.
Triple this to get the perimeter of the equilateral triangle.
3*4*sqrt(15) = 12*sqrt(15) cm
12*sqrt(15) = 46.4758002 approximately
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