The mid-points of sides AB and AC of a triangle ABC are respectively X and Y. If `(BC + XY) = 12` units, then the value of `(BC-XY)` is

Options

A. 2 units
B. 6 units
C. 8 units
D. 4 units

Answer

To solve the problem, we need to find the value of \( BC - XY \) given that \( BC + XY = 12 \).

### Step-by-Step Solution:

1. **Understanding the Midpoints**:
- Let \( X \) be the midpoint of side \( AB \) and \( Y \) be the midpoint of side \( AC \) in triangle \( ABC \).
- By the properties of midpoints, we know that \( XY \) (the line segment connecting the midpoints) is parallel to \( BC \) and half its length. Therefore, we can express \( XY \) in terms of \( BC \):
\[ XY = \frac{1}{2} BC \]

2. **Setting Up the Equation**:
- We are given that:
\[ BC + XY = 12 \]
- Substituting \( XY \) with \( \frac{1}{2} BC \):
\[ BC + \frac{1}{2} BC = 12 \]

3. **Combining Like Terms**:
- Combine the terms on the left side:
\[ \frac{3}{2} BC = 12 \]

4. **Solving for \( BC \)**:
- To isolate \( BC \), multiply both sides by \( \frac{2}{3} \):
\[ BC = 12 \times \frac{2}{3} = 8 \]

5. **Finding \( XY \)**:
- Now, substitute \( BC \) back into the equation for \( XY \):
\[ XY = \frac{1}{2} BC = \frac{1}{2} \times 8 = 4 \]

6. **Calculating \( BC - XY \)**:
- Finally, we can find \( BC - XY \):
\[ BC - XY = 8 - 4 = 4 \]

### Final Answer:
The value of \( BC - XY \) is \( 4 \) units.