Sure, let's solve this step by step.
1. Understanding the Sides Ratio: The sides of the triangle are given in the ratio [tex]\( 12: 17: 25 \)[/tex].
2. Given Perimeter: The perimeter of the triangle is given as 540 cm.
3. Calculate the Sum of the Ratios:
[tex]\[
12 + 17 + 25 = 54
\][/tex]
4. Determine the Actual Lengths of the Sides:
- The total ratio is 54.
- To find the actual length of each side, we multiply each ratio by the perimeter divided by the sum of the ratios.
Hence, the side lengths will be:
[tex]\[
a = \frac{540 \times 12}{54} = 120 \text{ cm}
\][/tex]
[tex]\[
b = \frac{540 \times 17}{54} = 170 \text{ cm}
\][/tex]
[tex]\[
c = \frac{540 \times 25}{54} = 250 \text{ cm}
\][/tex]
5. Calculate the Semi-Perimeter (s):
[tex]\[
s = \frac{540}{2} = 270 \text{ cm}
\][/tex]
6. Using Heron's Formula to Find the Area:
Heron's formula states that the area [tex]\( A \)[/tex] of a triangle whose sides are [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] is given by:
[tex]\[
A = \sqrt{s(s - a)(s - b)(s - c)}
\][/tex]
Substituting the values we get:
[tex]\[
A = \sqrt{270(270 - 120)(270 - 170)(270 - 250)}
\][/tex]
Simplifying inside the square root:
[tex]\[
A = \sqrt{270 \times 150 \times 100 \times 20}
\][/tex]
[tex]\[
A = \sqrt{81000000}
\][/tex]
[tex]\[
A = 9000 \text{ cm}^2
\][/tex]
Therefore, the area of the triangle is [tex]\( 9000 \)[/tex] square centimeters.
