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