The sides of a triangle having area 7776 sq. cm are in the ratio 3:4:5. The perimeter of the triangle is

Options

A. 432 cm
B. 400 cm
C. 412 cm
D. 424 cm

Answer

To find the perimeter of a triangle with an area of 7776 sq. cm and sides in the ratio 3:4:5, we can follow these steps:

### Step 1: Assign Variables to the Sides
Let the sides of the triangle be represented as:
- Side A = 3x
- Side B = 4x
- Side C = 5x

### Step 2: Use the Area Formula
The area of a triangle can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
In this case, we can consider Side B (4x) as the base and Side A (3x) as the height. Thus, we can write:
\[ \text{Area} = \frac{1}{2} \times 4x \times 3x = 6x^2 \]
Given that the area is 7776 sq. cm, we set up the equation:
\[ 6x^2 = 7776 \]

### Step 3: Solve for x
To find x, we first divide both sides by 6:
\[ x^2 = \frac{7776}{6} = 1296 \]
Now, take the square root of both sides:
\[ x = \sqrt{1296} = 36 \]

### Step 4: Calculate the Lengths of the Sides
Now that we have the value of x, we can find the lengths of the sides:
- Side A = 3x = 3(36) = 108 cm
- Side B = 4x = 4(36) = 144 cm
- Side C = 5x = 5(36) = 180 cm

### Step 5: Calculate the Perimeter
The perimeter of the triangle is the sum of all three sides:
\[ \text{Perimeter} = A + B + C = 108 + 144 + 180 \]
Calculating this gives:
\[ \text{Perimeter} = 432 \text{ cm} \]

### Final Answer
The perimeter of the triangle is **432 cm**.

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