For triangle ABC with vertices A(2, 4), B(3, 0) and C(5, 2) in the Cartesian plane, find the equation of the median AD, where D is the midpoint of side BC.

Introduction / Context:
This question is from coordinate geometry and focuses on the concept of a median of a triangle. A median is a line segment joining a vertex of a triangle to the midpoint of the opposite side. Here, you must find the equation of the median AD given the coordinates of the vertices of triangle ABC.

Given Data / Assumptions:

• A(2, 4), B(3, 0), C(5, 2) are the vertices of triangle ABC.
• D is the midpoint of BC.
• We need the equation of line AD in standard form.
• All coordinates lie in the usual Cartesian plane.

Concept / Approach:
To find the equation of a median, we first compute the midpoint of the opposite side using the midpoint formula. Then we find the equation of the line passing through the given vertex and this midpoint. The two main tools are:

• Midpoint formula: midpoint of (x1, y1) and (x2, y2) is ((x1 + x2) / 2, (y1 + y2) / 2).
• Two point form or slope form of a straight line equation.

Step-by-Step Solution:
Step 1: Identify the coordinates of B and C: B(3, 0), C(5, 2). Step 2: Compute the midpoint D of BC: D = ((3 + 5) / 2, (0 + 2) / 2) = (8 / 2, 2 / 2) = (4, 1). Step 3: We now need the equation of the line joining A(2, 4) and D(4, 1). Step 4: Calculate the slope m of AD: m = (1 − 4) / (4 − 2) = (−3) / 2 = −3 / 2. Step 5: Use the point-slope form: y − y1 = m(x − x1) with point A(2, 4). Step 6: Substitute: y − 4 = (−3 / 2)(x − 2). Step 7: Multiply both sides by 2 to clear the denominator: 2y − 8 = −3x + 6. Step 8: Rearrange to standard form: 3x + 2y − 14 = 0 or equivalently 3x + 2y = 14.

Verification / Alternative check:
Substitute A(2, 4) into 3x + 2y = 14: 3 * 2 + 2 * 4 = 6 + 8 = 14, which satisfies the equation. Substitute D(4, 1): 3 * 4 + 2 * 1 = 12 + 2 = 14, also satisfied. Because both A and D lie on the line, and AD is the median by construction, 3x + 2y = 14 is the correct equation.

Why Other Options Are Wrong:
3x − 2y = 14 and 3x − 2y = 2 do not pass through both A and D when tested by substitution.
3x + 2y = 2 is incorrect because substituting A or D does not satisfy the equation.
2x + 3y = 14 is a different line with a different slope and does not represent median AD.

Common Pitfalls:
Students sometimes make arithmetic mistakes in calculating the midpoint or the slope. Another common error is mishandling signs when rearranging the linear equation. Always verify your final equation by checking whether it passes through both the given vertex and the computed midpoint.

Final Answer:
The equation of the median AD is 3x + 2y = 14.