ABC is a triangle. D is a point on AB such that `AD = (1)/(4) AB and E` is a point an AC such that `AE = (1)/(4) AC`. Then `D E = `_______

Options

A. BC
B. 2 BC
C. `(1)/(4)` BC
D. `(1)/(2)` BC

Answer

To solve the problem, we need to find the length of segment \( DE \) in triangle \( ABC \) given the points \( D \) and \( E \) on sides \( AB \) and \( AC \) respectively.

### Step-by-Step Solution:

1. **Identify the Points**:
- Let \( A \), \( B \), and \( C \) be the vertices of triangle \( ABC \).
- Point \( D \) is on side \( AB \) such that \( AD = \frac{1}{4} AB \).
- Point \( E \) is on side \( AC \) such that \( AE = \frac{1}{4} AC \).

2. **Express Lengths**:
- Let \( AB = c \) and \( AC = b \).
- Then, \( AD = \frac{1}{4}c \) and \( AE = \frac{1}{4}b \).

3. **Use Similar Triangles**:
- Triangles \( ADE \) and \( ABC \) are similar because they share angle \( A \) and both have corresponding angles equal (angle \( DAE \) and angle \( BAC \)).
- Therefore, we can set up a ratio of corresponding sides:
\[ \frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC} \]

4. **Substituting Known Values**:
- From the similarity, we know:
\[ \frac{AD}{AB} = \frac{1/4 \cdot c}{c} = \frac{1}{4} \]
\[ \frac{AE}{AC} = \frac{1/4 \cdot b}{b} = \frac{1}{4} \]
- Thus, we can write:
\[ \frac{DE}{BC} = \frac{1}{4} \]

5. **Finding \( DE \)**:
- Rearranging gives us:
\[ DE = \frac{1}{4} BC \]

### Final Answer:
Thus, the length of segment \( DE \) is \( \frac{1}{4} BC \).