The area of a triangle is `216 cm^(2)` and its sides are in the ratio `3 : 4:5`. The perimeter of the triangle is :

Options

A. 6 cm
B. 12 cm
C. 36 cm
D. 72 cm

Answer

To find the perimeter of the triangle given that its area is \(216 \, \text{cm}^2\) and the sides are in the ratio \(3:4:5\), we can follow these steps:

### Step 1: Set up the sides of the triangle
Given the ratio of the sides is \(3:4:5\), we can express the sides as:
- Side 1 = \(3x\)
- Side 2 = \(4x\)
- Side 3 = \(5x\)

### Step 2: Use the area formula for a triangle
The area \(A\) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
For a right triangle (which this is, since \(3^2 + 4^2 = 5^2\)), we can take \(3x\) as the base and \(4x\) as the height. Thus, the area can be expressed as:
\[ A = \frac{1}{2} \times (3x) \times (4x) = \frac{12x^2}{2} = 6x^2 \]

### Step 3: Set the area equal to the given area
We know the area is \(216 \, \text{cm}^2\), so we set up the equation:
\[ 6x^2 = 216 \]

### Step 4: Solve for \(x^2\)
To find \(x^2\), divide both sides by \(6\):
\[ x^2 = \frac{216}{6} = 36 \]

### Step 5: Solve for \(x\)
Now take the square root of both sides:
\[ x = \sqrt{36} = 6 \]

### Step 6: Find the lengths of the sides
Now we can find the lengths of the sides:
- Side 1 = \(3x = 3 \times 6 = 18 \, \text{cm}\)
- Side 2 = \(4x = 4 \times 6 = 24 \, \text{cm}\)
- Side 3 = \(5x = 5 \times 6 = 30 \, \text{cm}\)

### Step 7: Calculate the perimeter
The perimeter \(P\) of the triangle is the sum of all its sides:
\[ P = 18 + 24 + 30 = 72 \, \text{cm} \]

### Final Answer
The perimeter of the triangle is \(72 \, \text{cm}\).

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