If `Delta_(1)` is the area of the triangle formed by the centroid and two vertices of a triangle `Delta_(2)` is the area of the triangle formed by the mid- point of the sides of the same triangle, then `Delta_(1):Delta_(2)`=

Options

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

Explanation

To solve the problem, we need to find the ratio of the areas of two triangles, \( \Delta_1 \) and \( \Delta_2 \), formed by specific points related to a triangle \( ABC \).

### Step-by-Step Solution

1. **Understanding the Triangles**:
- Let \( A \), \( B \), and \( C \) be the vertices of triangle \( ABC \).
- The centroid \( G \) of triangle \( ABC \) divides the triangle into three smaller triangles of equal area.
- The midpoints of sides \( AB \), \( BC \), and \( CA \) are denoted as \( M_{AB} \), \( M_{BC} \), and \( M_{CA} \) respectively.

2. **Area of Triangle Formed by Centroid and Two Vertices**:
- The area \( \Delta_1 \) is the area of triangle \( ABG \) (or any triangle formed by the centroid and two vertices).
- Since the centroid divides the triangle into three equal areas, we have:
\[ \Delta_1 = \frac{1}{3} \times \text{Area of } \triangle ABC = \frac{1}{3} \Delta \]
where \( \Delta \) is the area of triangle \( ABC \).

3. **Area of Triangle Formed by Midpoints**:
- The area \( \Delta_2 \) is the area of triangle \( M_{AB}M_{BC}M_{CA} \).
- The triangle formed by the midpoints of the sides of triangle \( ABC \) has an area that is \( \frac{1}{4} \) of the area of triangle \( ABC \):
\[ \Delta_2 = \frac{1}{4} \Delta \]

4. **Finding the Ratio \( \Delta_1 : \Delta_2 \)**:
- Now, we can find the ratio of \( \Delta_1 \) to \( \Delta_2 \):
\[ \frac{\Delta_1}{\Delta_2} = \frac{\frac{1}{3} \Delta}{\frac{1}{4} \Delta} \]
- The \( \Delta \) cancels out:
\[ \frac{\Delta_1}{\Delta_2} = \frac{\frac{1}{3}}{\frac{1}{4}} = \frac{1}{3} \times \frac{4}{1} = \frac{4}{3} \]

5. **Final Ratio**:
- Thus, the ratio \( \Delta_1 : \Delta_2 \) is:
\[ \Delta_1 : \Delta_2 = 4 : 3 \]

### Conclusion
The ratio \( \Delta_1 : \Delta_2 \) is \( 4 : 3 \).