X and Y are points on sides AB and AC respectively of `DeltaABC`. For which of the following cases will XY be parallel to BC?

Options

A. AX = 1.3 cm, XB = 3.9 cm,
AY = 2.8 cm, YC = 5.6 cm,
B. AX = 1.3 cm, XB = 3.9 cm,
AY = 2.8 cm, YC = 8.4 cm
C. AX = 1.3 cm, XB = 2.6 cm,
AY = 2.8 cm, YC = 8.4 cm
D. AX = 1.3 cm, XB = 2.6 cm,
AY = 2.8 cm, YC = 11.2 cm

Answer

To determine when line segment XY is parallel to line segment BC in triangle ABC, we can use the Basic Proportionality Theorem (also known as Thales' theorem). This theorem states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.

### Step-by-Step Solution:

1. **Identify the Points and Sides**:
- Let X be a point on side AB and Y be a point on side AC of triangle ABC.
- We need to find the conditions under which XY is parallel to BC.

2. **Set Up Ratios**:
- According to the Basic Proportionality Theorem, if XY is parallel to BC, then:
\[ \frac{AX}{XB} = \frac{AY}{YC} \]

3. **Evaluate Each Option**:
- We will evaluate the given options to see if they satisfy the condition above.

**Option A**:
- \( AX = 1.3, XB = 3.9, AY = 2.8, YC = 5.6 \)
- Calculate the ratios:
\[ \frac{AX}{XB} = \frac{1.3}{3.9} = \frac{1}{3} \quad \text{and} \quad \frac{AY}{YC} = \frac{2.8}{5.6} = \frac{1}{2} \]
- Since \(\frac{1}{3} \neq \frac{1}{2}\), this option is incorrect.

**Option B**:
- \( AX = 1.3, XB = 3.9, AY = 2.8, YC = 8.4 \)
- Calculate the ratios:
\[ \frac{AX}{XB} = \frac{1.3}{3.9} = \frac{1}{3} \quad \text{and} \quad \frac{AY}{YC} = \frac{2.8}{8.4} = \frac{1}{3} \]
- Since \(\frac{1}{3} = \frac{1}{3}\), this option is correct.

**Option C**:
- \( AX = 1.3, XB = 2.6, AY = 2.8, YC = 8.4 \)
- Calculate the ratios:
\[ \frac{AX}{XB} = \frac{1.3}{2.6} = \frac{1}{2} \quad \text{and} \quad \frac{AY}{YC} = \frac{2.8}{8.4} = \frac{1}{3} \]
- Since \(\frac{1}{2} \neq \frac{1}{3}\), this option is incorrect.

**Option D**:
- \( AX = 1.3, XB = 2.6, AY = 2.8, YC = 11.2 \)
- Calculate the ratios:
\[ \frac{AX}{XB} = \frac{1.3}{2.6} = \frac{1}{2} \quad \text{and} \quad \frac{AY}{YC} = \frac{2.8}{11.2} = \frac{1}{4} \]
- Since \(\frac{1}{2} \neq \frac{1}{4}\), this option is incorrect.

4. **Conclusion**:
- The only option that satisfies the condition for XY to be parallel to BC is **Option B**.