Let ABC be a given triangle. Points D and E are on sides AB and AC respectively and point F is on line segment DE. Let `(AD)/(AB)=x, (AE)/(AC)=y, (DF)/(DE)=z`. Let area of `DeltaBDF=Delta_(1)`, Area of `DeltaCEF=Delta_(2) ` and area of `DeltaABC=Delta`.
Q. `(Delta_(2))/(Delta)` is equal to :

Options

A. `(1-x)y(1-z)`
B. `(1-x)(1-y)z`
C. `x(1-y)(1-z)`
D. `(1-x)yz`

Answer

To solve the problem, we need to find the ratio of the area of triangle CEF (denoted as Δ2) to the area of triangle ABC (denoted as Δ). Given the relationships between the segments of the triangle, we can use the properties of similar triangles and the area formula for triangles.

### Step-by-Step Solution:

1. **Understanding the Ratios**:
We are given:
- \( \frac{AD}{AB} = x \)
- \( \frac{AE}{AC} = y \)
- \( \frac{DF}{DE} = z \)

2. **Area of Triangle ABC**:
The area of triangle ABC can be expressed as:
\[ \Delta = \frac{1}{2} \times AB \times AC \times \sin(\theta) \]
where \( \theta \) is the angle at vertex A.

3. **Finding Area of Triangle CEF**:
The area of triangle CEF can be expressed in terms of the segments:
\[ \Delta_2 = \frac{1}{2} \times CE \times CF \times \sin(\alpha) \]
where \( \alpha \) is the angle at vertex C.

4. **Using Ratios**:
From the given ratios, we can express CE and CF in terms of x, y, and z:
- \( CE = (1 - y) \cdot AC \)
- \( CF = (1 - z) \cdot DE \)

5. **Expressing DE**:
Since \( DE \) can be expressed in terms of \( AB \) and \( AC \):
\[ DE = (1 - x) \cdot AB \]

6. **Substituting Values**:
Substitute \( CE \) and \( CF \) into the area formula for Δ2:
\[ \Delta_2 = \frac{1}{2} \cdot (1 - y) \cdot AC \cdot (1 - z) \cdot DE \cdot \sin(\alpha) \]

7. **Finding the Ratio**:
Now we can find the ratio \( \frac{\Delta_2}{\Delta} \):
\[ \frac{\Delta_2}{\Delta} = \frac{(1 - y)(1 - z) \cdot DE \cdot \sin(\alpha)}{AB \cdot AC \cdot \sin(\theta)} \]

8. **Final Expression**:
Using the relationships:
\[ \frac{\Delta_2}{\Delta} = \left(\frac{AD}{AB}\right) \cdot \left(1 - \frac{AE}{AC}\right) \cdot \left(1 - \frac{DF}{DE}\right) \]
We can substitute \( x \), \( y \), and \( z \):
\[ \frac{\Delta_2}{\Delta} = x \cdot (1 - y) \cdot (1 - z) \]

### Final Answer:
Thus, the ratio \( \frac{\Delta_2}{\Delta} \) is equal to:
\[ \frac{\Delta_2}{\Delta} = x \cdot (1 - y) \cdot (1 - z) \]