Answer :
Given that the sides of the triangle are in the ratio 5:4:3 and its area is 54 cm², we need to find the perimeter of the triangle.
Let the sides of the triangle be \( 5x \), \( 4x \), and \( 3x \).
To find the perimeter, we need to determine the value of \( x \).
First, we can use Heron's formula to relate the area of the triangle to its sides. Heron's formula states that the area \( A \) of a triangle with sides \( a \), \( b \), and \( c \) is given by:
\[
A = \sqrt{s(s-a)(s-b)(s-c)}
\]
where \( s \) is the semi-perimeter of the triangle:
\[
s = \frac{a + b + c}{2}
\]
Substitute \( a = 5x \), \( b = 4x \), and \( c = 3x \):
\[
s = \frac{5x + 4x + 3x}{2} = \frac{12x}{2} = 6x
\]
Given that the area \( A \) is 54 cm², we can substitute into Heron's formula:
\[
54 = \sqrt{6x(6x - 5x)(6x - 4x)(6x - 3x)}
\]
Simplify the terms inside the square root:
\[
54 = \sqrt{6x \cdot x \cdot 2x \cdot 3x}
\]
\[
54 = \sqrt{36x^4}
\]
\[
54 = 6x^2
\]
Solve for \( x^2 \):
\[
54 = 6x^2
\]
\[
x^2 = \frac{54}{6} = 9
\]
\[
x = \sqrt{9} = 3
\]
Now that we have \( x = 3 \), we can find the lengths of the sides of the triangle:
\[
5x = 5 \times 3 = 15 \, \text{cm}
\]
\[
4x = 4 \times 3 = 12 \, \text{cm}
\]
\[
3x = 3 \times 3 = 9 \, \text{cm}
\]
The perimeter \( P \) of the triangle is the sum of its sides:
\[
P = 15 + 12 + 9 = 36 \, \text{cm}
\]
Thus, the perimeter of the triangle is \( 36 \, \text{cm} \).
Let the sides of the triangle be \( 5x \), \( 4x \), and \( 3x \).
To find the perimeter, we need to determine the value of \( x \).
First, we can use Heron's formula to relate the area of the triangle to its sides. Heron's formula states that the area \( A \) of a triangle with sides \( a \), \( b \), and \( c \) is given by:
\[
A = \sqrt{s(s-a)(s-b)(s-c)}
\]
where \( s \) is the semi-perimeter of the triangle:
\[
s = \frac{a + b + c}{2}
\]
Substitute \( a = 5x \), \( b = 4x \), and \( c = 3x \):
\[
s = \frac{5x + 4x + 3x}{2} = \frac{12x}{2} = 6x
\]
Given that the area \( A \) is 54 cm², we can substitute into Heron's formula:
\[
54 = \sqrt{6x(6x - 5x)(6x - 4x)(6x - 3x)}
\]
Simplify the terms inside the square root:
\[
54 = \sqrt{6x \cdot x \cdot 2x \cdot 3x}
\]
\[
54 = \sqrt{36x^4}
\]
\[
54 = 6x^2
\]
Solve for \( x^2 \):
\[
54 = 6x^2
\]
\[
x^2 = \frac{54}{6} = 9
\]
\[
x = \sqrt{9} = 3
\]
Now that we have \( x = 3 \), we can find the lengths of the sides of the triangle:
\[
5x = 5 \times 3 = 15 \, \text{cm}
\]
\[
4x = 4 \times 3 = 12 \, \text{cm}
\]
\[
3x = 3 \times 3 = 9 \, \text{cm}
\]
The perimeter \( P \) of the triangle is the sum of its sides:
\[
P = 15 + 12 + 9 = 36 \, \text{cm}
\]
Thus, the perimeter of the triangle is \( 36 \, \text{cm} \).