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 \( BC + XY = 12 \) units.

### Step-by-Step Solution:

1. **Understand the Midpoint Theorem**:
According to the midpoint theorem, the line segment joining the midpoints of two sides of a triangle is parallel to the third side and is half as long as the third side. In this case, since \( X \) and \( Y \) are the midpoints of sides \( AB \) and \( AC \) respectively, we can say:
\[ XY = \frac{1}{2} BC \]

2. **Set Up the Given Equation**:
We are given that:
\[ BC + XY = 12 \]

3. **Substitute the Midpoint Theorem into the Equation**:
From the midpoint theorem, we can substitute \( XY \) in the equation:
\[ BC + \frac{1}{2} BC = 12 \]

4. **Combine Like Terms**:
Combine \( BC \) and \( \frac{1}{2} BC \):
\[ \frac{3}{2} BC = 12 \]

5. **Solve for \( BC \)**:
To find \( BC \), multiply both sides by \( \frac{2}{3} \):
\[ BC = 12 \times \frac{2}{3} = 8 \]

6. **Find \( XY \)**:
Now, using the relationship \( XY = \frac{1}{2} BC \):
\[ XY = \frac{1}{2} \times 8 = 4 \]

7. **Calculate \( BC - XY \)**:
Now we can find \( BC - XY \):
\[ BC - XY = 8 - 4 = 4 \]

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