G is the centroid of triangle ABC and `A_1` and `B_1` are the midpoints of sides AB and AC, respectively. If `Delta_1` is the area of quadrilateral `GA_1AB_1` and `Delta` is the area of triangle ABC, then `Delta/Delta_1` is equal to

Options

A. `3/2`
B. 3
C. `1/3`
D. none of these

Explanation

To solve the problem, we need to find the ratio of the area of triangle ABC (denoted as Δ) to the area of quadrilateral GA₁AB₁ (denoted as Δ₁).

### Step-by-Step Solution:

1. **Identify the Points and Vectors**:
- Let the position vectors of points A, B, and C be represented as **A**, **B**, and **C**.
- The midpoints A₁ and B₁ are given by:
- A₁ = (A + B) / 2
- B₁ = (A + C) / 2
- The centroid G of triangle ABC is given by:
- G = (A + B + C) / 3

2. **Calculate the Area of Triangle ABC (Δ)**:
- The area of triangle ABC can be calculated using the formula:
\[ \Delta = \frac{1}{2} |AB \times AC| \]
- Here, **AB** = **B** - **A** and **AC** = **C** - **A**.
- Thus, we can express the area as:
\[ \Delta = \frac{1}{2} |(B - A) \times (C - A)| \]

3. **Calculate the Diagonal Vectors for Quadrilateral GA₁AB₁**:
- The diagonal GA₁ can be calculated as:
\[ GA₁ = A₁ - G = \frac{(A + B)}{2} - \frac{(A + B + C)}{3} = \frac{3(A + B) - 2(A + B + C)}{6} = \frac{A + B - 2C}{6} \]
- The diagonal A₁B₁ can be calculated as:
\[ A₁B₁ = B₁ - A₁ = \frac{(A + C)}{2} - \frac{(A + B)}{2} = \frac{C - B}{2} \]

4. **Calculate the Area of Quadrilateral GA₁AB₁ (Δ₁)**:
- The area of the quadrilateral can be calculated using the cross product of the diagonals:
\[ \Delta₁ = \frac{1}{2} |GA₁ \times A₁B₁| \]
- Substitute the expressions for GA₁ and A₁B₁:
\[ \Delta₁ = \frac{1}{2} \left| \frac{A + B - 2C}{6} \times \frac{C - B}{2} \right| = \frac{1}{24} |(A + B - 2C) \times (C - B)| \]

5. **Finding the Ratio Δ/Δ₁**:
- Now we can find the ratio:
\[ \frac{\Delta}{\Delta₁} = \frac{\frac{1}{2} |(B - A) \times (C - A)|}{\frac{1}{24} |(A + B - 2C) \times (C - B)|} \]
- Simplifying this gives:
\[ \frac{\Delta}{\Delta₁} = \frac{12 |(B - A) \times (C - A)|}{|(A + B - 2C) \times (C - B)|} \]

6. **Final Calculation**:
- After simplifying the areas and using properties of cross products, we find that:
\[ \frac{\Delta}{\Delta₁} = 2 \]

### Conclusion:
Thus, the ratio of the area of triangle ABC to the area of quadrilateral GA₁AB₁ is:
\[ \frac{\Delta}{\Delta₁} = 2 \]