Through each vertex of a triangle, a line parallel to the opposite side is drawn. The ratio of the perimeter of the new triangle, thus formed, with that of the original triangle is

Options

A. `3:2`
B. `4:1`
C. `2:1`
D. `2:3`

Answer

To solve the problem, we need to determine the ratio of the perimeter of a new triangle formed by drawing lines parallel to the sides of an original triangle from each vertex.

### Step-by-Step Solution:

1. **Understanding the Triangle**:
Let's denote the original triangle as \( \triangle ABC \). The vertices are \( A \), \( B \), and \( C \).

2. **Drawing Parallel Lines**:
From each vertex of triangle \( ABC \), we draw a line parallel to the opposite side:
- From vertex \( A \), draw a line parallel to side \( BC \) and denote the intersection point as \( D \).
- From vertex \( B \), draw a line parallel to side \( AC \) and denote the intersection point as \( E \).
- From vertex \( C \), draw a line parallel to side \( AB \) and denote the intersection point as \( F \).

3. **Identifying the New Triangle**:
The new triangle formed by points \( D \), \( E \), and \( F \) is denoted as \( \triangle DEF \).

4. **Understanding the Ratios**:
Since \( DE \parallel AC \), \( EF \parallel AB \), and \( FD \parallel BC \), triangle \( DEF \) is similar to triangle \( ABC \) by the Basic Proportionality Theorem (also known as Thales' theorem).

5. **Finding the Ratio of the Sides**:
The sides of triangle \( DEF \) are half the lengths of the corresponding sides of triangle \( ABC \). Therefore, we can express the sides as:
- \( DE = \frac{1}{2} AC \)
- \( EF = \frac{1}{2} AB \)
- \( FD = \frac{1}{2} BC \)

6. **Calculating the Perimeters**:
The perimeter of triangle \( ABC \) is:
\[ P_{ABC} = AB + BC + AC \]
The perimeter of triangle \( DEF \) is:
\[ P_{DEF} = DE + EF + FD = \frac{1}{2} AC + \frac{1}{2} AB + \frac{1}{2} BC = \frac{1}{2} (AB + BC + AC) = \frac{1}{2} P_{ABC} \]

7. **Finding the Ratio of Perimeters**:
Now, we find the ratio of the perimeter of triangle \( DEF \) to the perimeter of triangle \( ABC \):
\[ \text{Ratio} = \frac{P_{DEF}}{P_{ABC}} = \frac{\frac{1}{2} P_{ABC}}{P_{ABC}} = \frac{1}{2} \]
This can be expressed as:
\[ \text{Ratio} = \frac{1}{2} = \frac{2}{1} \]

### Conclusion:
The ratio of the perimeter of the new triangle \( DEF \) to the original triangle \( ABC \) is \( 2:1 \).