The perimeter of a triangle is 547.2 feet and the sides are in the ratio of 42: 45: 27. find the area of the triangle.

The answer is 12,782.5 feet².

The perimeter is the sum of all sides of a polygon. The perimeter (P) of the triangle is:
P = a + b + c

It is given:
P = 547.2 = a + b + c
a : b : c = 42 : 45 : 27 

Let's assume that sides are as the ratio:
a1 = 42,     b1 = 45,      c1 = 27
The perimeter will be: P1 = 42 + 45 + 27 = 114 feet

Now let's see how many times our triangle is bigger that this supposed one:
P ÷ P1 = 547.2 ÷ 114 = 4.8

This means that sides are 4.8 times bigger:
a = 42 · 4.8 = 201.6 feet
b = 45 · 4.8 = 216 feet
c = 27 · 4.8 = 129.6 feet

Let's just check it out: P = 201.6 + 216 + 129.6 = 547.2 feet

Now, let's calculate the area of the triangle using the Heron's formula:
A = √(s(s - a)(s - b)(s - c))
s = (a + b + c)/2 = P/2 = 547.2/2 = 273.6 feet

A =√(273.6 · (273.6 - 201.6) · (273.6 - 216) · (273.6 - 129.6))
   = √(273.6 · 72 · 57.6 · 144) = √163,369,044.5 = 12,782.5 feet²

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