If the sides of an equilateral triangle are increased by 20%, 30% and 50% respectively to form a new triangle, what is the percentage increase in the perimeter of the equilateral triangle ?

Answer

To solve the problem of finding the percentage increase in the perimeter of an equilateral triangle when its sides are increased by 20%, 30%, and 50%, we can follow these steps:

### Step 1: Define the original side length
Let the original side length of the equilateral triangle be \( x \) cm.

### Step 2: Calculate the new lengths of the sides
- The first side is increased by 20%:
\[ \text{New length of side 1} = x + 0.20x = 1.2x \]

- The second side is increased by 30%:
\[ \text{New length of side 2} = x + 0.30x = 1.3x \]

- The third side is increased by 50%:
\[ \text{New length of side 3} = x + 0.50x = 1.5x \]

### Step 3: Calculate the new perimeter
The new perimeter \( P' \) of the triangle can be calculated by adding the new lengths of the sides:
\[ P' = 1.2x + 1.3x + 1.5x \]
Combining the terms:
\[ P' = (1.2 + 1.3 + 1.5)x = 4.0x \]

### Step 4: Calculate the original perimeter
The original perimeter \( P \) of the equilateral triangle is:
\[ P = 3x \]

### Step 5: Calculate the increase in perimeter
The increase in perimeter is:
\[ \text{Increase} = P' - P = 4.0x - 3x = 1.0x \]

### Step 6: Calculate the percentage increase
The percentage increase in perimeter can be calculated using the formula:
\[ \text{Percentage Increase} = \left(\frac{\text{Increase}}{P}\right) \times 100 \]
Substituting the values:
\[ \text{Percentage Increase} = \left(\frac{1.0x}{3x}\right) \times 100 = \frac{1.0}{3} \times 100 = \frac{100}{3} \approx 33.33\% \]

### Final Answer
The percentage increase in the perimeter of the equilateral triangle is \( \frac{100}{3}\% \) or \( 33 \frac{1}{3}\% \).

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