For triangle ABC with vertices A(2, 4), B(3, 0) and C(5, 2), what is the equation of the median AD drawn from vertex A to side BC?

Introduction / Context:
This coordinate geometry question tests your understanding of medians in a triangle and the method of finding equations of lines through two points. A median from a vertex is a line drawn from that vertex to the midpoint of the opposite side. Once we find the midpoint of side BC, we can determine the equation of the line passing through A and this midpoint.

Given Data / Assumptions:

• Vertices of triangle ABC: A(2, 4), B(3, 0), and C(5, 2).
• We need the equation of median AD, where D is the midpoint of segment BC.
• We use standard formulas for midpoint and line equation.

Concept / Approach:
The median from A goes to the midpoint of BC. So the first step is to find the midpoint D of points B and C. Then, using the coordinates of A and D, we find the slope of line AD. With the slope and one point, we can write the equation in point slope form, and then convert to the standard form ax + by = c using integer coefficients, which can be matched against the given options.

Step-by-Step Solution:

Verification / Alternative check:
Check that both A and D satisfy 3x + 2y = 14. For A(2, 4): 3*2 + 2*4 = 6 + 8 = 14, correct. For D(4, 1): 3*4 + 2*1 = 12 + 2 = 14, also correct. Since the line through A and D is unique, this equation correctly represents the median AD.

Why Other Options Are Wrong:

Common Pitfalls:
Common errors include computing the midpoint incorrectly or reversing the direction when calculating the slope. Another pitfall is forgetting to distribute the negative sign when expanding -3(x - 2). Being careful with arithmetic at each step ensures the final line equation is accurate.

Final Answer:
3x + 2y = 14