In triangle ABC, X and Y are the midpoints of sides AB and AC respectively. Segment XY is the line joining these midpoints and is parallel to side BC. If BC + XY = 12 units, what is the length of side BC (in units)?

Introduction / Context:
This problem involves the mid segment of a triangle, which is a standard concept in geometry. The mid segment theorem states that the segment joining the midpoints of two sides of a triangle is parallel to the third side and has half its length. Here that property is combined with a simple linear equation involving BC and XY.

Given Data / Assumptions:
- Triangle ABC is given.
- X is the midpoint of AB and Y is the midpoint of AC.
- Segment XY joins these midpoints, so XY ∥ BC by the mid segment theorem.
- It is given that BC + XY = 12 units.
- We need to find the length of BC.

Concept / Approach:
By the mid segment theorem:
XY = (1 / 2) * BC.
The condition BC + XY = 12 gives a simple equation involving BC and XY. Substituting XY in terms of BC into this equation allows us to solve for BC directly. Once BC is known, XY can also be found if needed, but the question only asks for BC.

Step-by-Step Solution:
Step 1: By the midpoint or mid segment theorem, XY = (1 / 2) * BC. Step 2: Use the given relation BC + XY = 12. Step 3: Substitute XY from Step 1 into this relation: BC + (1 / 2) * BC = 12. Step 4: Combine like terms: BC + (1 / 2) * BC = (3 / 2) * BC. Step 5: So (3 / 2) * BC = 12. Step 6: Solve for BC by multiplying both sides by 2 / 3: BC = 12 * (2 / 3) = 24 / 3 = 8 units.

Verification / Alternative Check:
If BC = 8 units, then XY = (1 / 2) * 8 = 4 units. Now check the given condition: BC + XY = 8 + 4 = 12 units, which matches the information in the question. Hence BC = 8 units is consistent and correct.

Why Other Options Are Wrong:
Option a, 10 units, would give XY = 5 units and BC + XY = 10 + 5 = 15, not 12.
Option c, 6 units, would give XY = 3 units, and BC + XY = 6 + 3 = 9, which again does not match 12.
Option d, 4 units, would give XY = 2 units, and BC + XY = 4 + 2 = 6, still not equal to 12.

Common Pitfalls:
Some students mistakenly think the mid segment equals the third side rather than half of it, leading to BC = XY or BC − XY = 12 instead of BC + XY = 12. Others may forget that both BC and XY must be positive, which restricts possible solutions. Carefully applying the mid segment theorem and forming the correct equation avoids these issues.

Final Answer:
The length of side BC is 8 units.