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

Introduction / Context:
This geometry problem makes use of the mid point theorem in triangles. When a segment joins the mid points of two sides of a triangle, it has a special relationship with the third side. Here, that segment is XY, and the third side is BC. You are given the sum BC + XY and asked to compute BC − XY, which requires understanding the proportional relation between BC and XY and then using simple algebra.

Given Data / Assumptions:

• ABC is a triangle.
• X is the mid point of side AB.
• Y is the mid point of side AC.
• Segment XY is drawn joining X and Y.
• BC + XY = 12 units.
• We must find BC − XY in units.

Concept / Approach:
The mid point theorem states that the segment joining the mid points of two sides of a triangle is parallel to the third side and has length equal to half of that third side. So in triangle ABC, XY is parallel to BC and XY = (1 / 2) * BC. Using this relationship, we can express XY in terms of BC and substitute into the equation BC + XY = 12. Solving this simple equation gives the value of BC, and then XY can be found and subtracted to get BC − XY.

Step-by-Step Solution:
From the mid point theorem, XY is parallel to BC and XY = (1 / 2) * BC.Let BC = b. Then XY = b / 2.Given BC + XY = 12, substitute: b + (b / 2) = 12.Combine like terms: (3b / 2) = 12.Solve for b: b = (2 / 3) * 12 = 8 units.Then XY = b / 2 = 8 / 2 = 4 units, so BC − XY = 8 − 4 = 4 units.

Verification / Alternative check:
Check that BC and XY satisfy the original condition. With BC = 8 units and XY = 4 units, BC + XY = 8 + 4 = 12 units, which matches the given sum. Also, XY = 4 is indeed half of BC = 8, confirming that the mid point theorem relation is satisfied. Finally, BC − XY = 8 − 4 = 4 units, so the computed value is consistent with all conditions in the problem.

Why Other Options Are Wrong:
Option 8 units would mean BC − XY equals the same value as BC, which is impossible unless XY were zero, contradicting the mid point theorem. Option 6 units would correspond to some incorrect algebra leading to a different value of BC. Option 2 units and 10 units similarly do not satisfy both the sum condition and the mid segment relation XY = BC / 2. Only 4 units works when you test it back in the original equation.

Common Pitfalls:
Some students incorrectly assume XY = BC rather than BC / 2. Others might misinterpret BC + XY as BC * XY or set up the equation with reversed roles. Mistakes in solving (3b / 2) = 12, such as multiplying by 2 and dividing incorrectly, can also occur. Remembering the exact statement of the mid point theorem and doing the algebra step by step avoids these issues.

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