The ratio of sides of a triangle is `3:4:5`. If area of the triangle is 72 square unit, then the length of the smallest side is :

Options

A. `4sqrt(3)` unit
B. `5sqrt(3)` unit
C. `6sqrt(3)` unit
D. `3sqrt(3)` unit

Answer

To find the length of the smallest side of the triangle given the ratio of its sides and its area, we can follow these steps:

### Step 1: Define the sides of the triangle
Given the ratio of the sides of the triangle is \(3:4:5\), we can express the sides in terms of a variable \(x\):
- Let the sides be:
- \(a = 3x\)
- \(b = 4x\)
- \(c = 5x\)

### Step 2: Calculate the semi-perimeter (s)
The semi-perimeter \(s\) of the triangle is calculated using the formula:
\[ s = \frac{a + b + c}{2} \]
Substituting the values of \(a\), \(b\), and \(c\):
\[ s = \frac{3x + 4x + 5x}{2} = \frac{12x}{2} = 6x \]

### Step 3: Use Heron's formula to find the area
Heron's formula for the area \(A\) of a triangle is given by:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values we have:
\[ A = \sqrt{6x(6x - 3x)(6x - 4x)(6x - 5x)} \]
This simplifies to:
\[ A = \sqrt{6x(3x)(2x)(x)} = \sqrt{36x^4} = 6x^2 \]

### Step 4: Set the area equal to the given value
We know from the problem that the area of the triangle is \(72\) square units. Therefore, we set up the equation:
\[ 6x^2 = 72 \]

### Step 5: Solve for \(x\)
To find \(x\), we divide both sides by \(6\):
\[ x^2 = \frac{72}{6} = 12 \]
Taking the square root of both sides, we get:
\[ x = \sqrt{12} = 2\sqrt{3} \]

### Step 6: Find the length of the smallest side
The smallest side of the triangle is \(a = 3x\):
\[ a = 3(2\sqrt{3}) = 6\sqrt{3} \]

### Conclusion
Thus, the length of the smallest side of the triangle is \(6\sqrt{3}\).

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