In which of the following options ,DE ||AB where D and E lie on the sides AC and BC of `DeltaABC` respectively ?

Options

A. AD = 6 , EC = 14 , BC = 18 , DC = 21
B. BE = 20 , Dc = 10 , AC = 25 , BC =36
C. AC = 10 , Dc = 4 , EC =2 , BC =6
D. None of these

Answer

To determine in which of the given options DE is parallel to AB in triangle ABC, we will use the properties of similar triangles and 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 Triangle and Points**:
Let triangle ABC be given with points D and E on sides AC and BC respectively.

2. **Understand the Condition for Parallel Lines**:
According to the Basic Proportionality Theorem, if DE || AB, then:
\[ \frac{AD}{DC} = \frac{BE}{EC} \quad \text{or} \quad \frac{AD}{AB} = \frac{BE}{BC} \]

3. **Check Each Option**:
- **Option A**:
- Given: \( AD = 6 \), \( EC = 14 \), \( BC = 18 \)
- Calculate \( EB = BC - EC = 18 - 14 = 4 \)
- Calculate \( DC = 20 \)
- Check the ratio:
\[ \frac{AD}{DC} = \frac{6}{20} = \frac{3}{10} \]
\[ \frac{BE}{EC} = \frac{4}{14} = \frac{2}{7} \]
- Since \( \frac{3}{10} \neq \frac{2}{7} \), this option does not satisfy the condition.

- **Option B**:
- Given: \( BE = 20 \), \( DC = 10 \), \( AC = 25 \)
- Calculate \( AD = AC - DC = 25 - 10 = 15 \)
- Calculate \( CE = BC - BE = 36 - 20 = 16 \)
- Check the ratio:
\[ \frac{AD}{DC} = \frac{15}{10} = \frac{3}{2} \]
\[ \frac{BE}{EC} = \frac{20}{16} = \frac{5}{4} \]
- Since \( \frac{3}{2} \neq \frac{5}{4} \), this option does not satisfy the condition.

- **Option C**:
- Given: \( AC = 10 \), \( DC = 4 \), \( CE = 2 \), \( BC = 6 \)
- Calculate \( AD = AC - DC = 10 - 4 = 6 \)
- Calculate \( BE = BC - CE = 6 - 2 = 4 \)
- Check the ratio:
\[ \frac{AD}{DC} = \frac{6}{4} = \frac{3}{2} \]
\[ \frac{BE}{EC} = \frac{4}{2} = 2 \]
- Since \( \frac{3}{2} \neq 2 \), this option does not satisfy the condition.

4. **Conclusion**:
After checking all options, only **Option A** satisfies the condition for DE to be parallel to AB.