If each sides of a triangle is doubled then find the ratio of the area of the new triangle thus formed and the given triangle.

Options

A. `4 : 1`
B. `5 : 1`
C. `6 : 1`
D. `7 : 1`

Explanation

To solve the problem of finding the ratio of the area of a new triangle formed when each side of a triangle is doubled, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a triangle with sides \( A \), \( B \), and \( C \). When each side of this triangle is doubled, the new sides become \( 2A \), \( 2B \), and \( 2C \).

2. **Using Heron's Formula**:
The area \( A \) of a triangle can be calculated using Heron's formula:
\[ \text{Area} = \sqrt{s(s - A)(s - B)(s - C)} \]
where \( s \) is the semi-perimeter given by:
\[ s = \frac{A + B + C}{2} \]

3. **Calculating the Semi-Perimeter of the New Triangle**:
For the new triangle with sides \( 2A \), \( 2B \), and \( 2C \), the semi-perimeter \( s' \) is:
\[ s' = \frac{2A + 2B + 2C}{2} = A + B + C = 2s \]

4. **Calculating the Area of the New Triangle**:
Using Heron's formula for the new triangle:
\[ \text{Area}_{\text{new}} = \sqrt{s'(s' - 2A)(s' - 2B)(s' - 2C)} \]
Substituting \( s' = 2s \):
\[ \text{Area}_{\text{new}} = \sqrt{2s(2s - 2A)(2s - 2B)(2s - 2C)} \]
This simplifies to:
\[ \text{Area}_{\text{new}} = \sqrt{2s \cdot 2(s - A) \cdot 2(s - B) \cdot 2(s - C)} = \sqrt{16s(s - A)(s - B)(s - C)} \]
Thus:
\[ \text{Area}_{\text{new}} = 4 \sqrt{s(s - A)(s - B)(s - C)} = 4 \cdot \text{Area}_{\text{given}} \]

5. **Finding the Ratio of Areas**:
Therefore, the ratio of the area of the new triangle to the area of the original triangle is:
\[ \frac{\text{Area}_{\text{new}}}{\text{Area}_{\text{given}}} = \frac{4 \cdot \text{Area}_{\text{given}}}{\text{Area}_{\text{given}}} = 4 \]
Hence, the ratio is:
\[ \text{Area}_{\text{new}} : \text{Area}_{\text{given}} = 4 : 1 \]

### Final Answer:
The ratio of the area of the new triangle to the area of the given triangle is \( 4 : 1 \).