In triangle ABC, points X and Y are the midpoints of sides AB and AC respectively. If BC + XY = 12 units, then using the mid-segment (midline) theorem, what is the value of BC − XY (in units)?

Introduction / Context:
This question tests the mid-segment (midline) theorem in triangles. When you join the midpoints of two sides of a triangle, you create a segment that is parallel to the third side and exactly half its length. Using this property, we can relate BC and XY and then solve a simple algebraic equation.

Given Data / Assumptions:

• Triangle ABC is any triangle.
• X is the midpoint of AB.
• Y is the midpoint of AC.
• Segment XY is the line joining these midpoints.
• BC + XY = 12 units.
• We must find BC − XY.

Concept / Approach:
The mid-segment theorem states: XY ∥ BC and XY = (1/2) * BC So if we let BC = b, then XY = b / 2. Given BC + XY in terms of b, we can solve for b, and then compute BC − XY using the same expressions.

Step-by-Step Solution:
Let BC = b. Then XY = b / 2 (by the mid-segment theorem). We are told BC + XY = 12, so b + (b / 2) = 12. Combine terms: (3b / 2) = 12. Multiply both sides by 2: 3b = 24, so b = 8. Then XY = b / 2 = 8 / 2 = 4. Now BC − XY = 8 − 4 = 4 units.

Verification / Alternative check:
Check the given condition with b = 8 and XY = 4. Then BC + XY = 8 + 4 = 12, which matches the question, confirming the algebra and the derived value of BC − XY.

Why Other Options Are Wrong:
8 units would arise if you misinterpreted XY as equal to BC, not half. 6 units and 2 units come from incorrect algebra or from choosing wrong expressions for XY in terms of BC.

Common Pitfalls:
Students sometimes forget that the mid-segment is half the corresponding side, or they reverse the fraction and take XY = 2 * BC. Others may incorrectly add or subtract when solving the equation b + b/2 = 12. Carefully write the relation and combine like terms to avoid such mistakes.

Final Answer:
The value of BC − XY is 4 units.