The area of a triangle is $216\text{ cm}^2$ and its sides are in the ratio $3 : 4 : 5$. The perimeter of the triangle is

### Concept & Ratio Properties When the sides of a triangle are in the ratio $3 : 4 : 5$, they form a Pythagorean triplet (since $3^2 + 4^2 = 5^2$). This means the triangle is a right-angled triangle. The area of a right-angled triangle is given by: $$ \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} $$ ### Step-by-Step Solution 1. Let the sides of the triangle be $3x$, $4x$, and $5x$. 2. The base and height are the legs of the right triangle: $3x$ and $4x$. 3. Substitute the values into the area formula: $216 = \frac{1}{2} \times 3x \times 4x$ 4. Simplify the equation: $216 = 6x^2$ $x^2 = 36$ $x = 6$ 5. The sides of the triangle are $3(6) = 18\text{ cm}$, $4(6) = 24\text{ cm}$, and $5(6) = 30\text{ cm}$. 6. Calculate the perimeter: $\text{Perimeter} = 18 + 24 + 30 = 72\text{ cm}$. ### Exam Strategy & Shortcut Instead of calculating individual sides, note that the perimeter in terms of $x$ is $3x + 4x + 5x = 12x$. Once you find $x = 6$ from the area equation $6x^2 = 216$, directly multiply $12 \times 6 = 72$. This saves calculation time. ### Common Pitfall A common mistake is assuming the sides are exactly 3, 4, and 5 and just finding the perimeter of that basic triangle ($12\text{ cm}$). Always use a variable (like $x$) to represent the common ratio multiplier. ### Final Answer Therefore, the correct answer is **72 cm**.