In coordinate geometry, triangle ABC has vertices A(-5, 4), B(-4, 0), and C(-2, 2). Find the equation of the median AD, where D is the midpoint of side BC and the median is drawn from vertex A to side BC.

Introduction / Context:
This coordinate geometry question asks you to find the equation of a median of a triangle. A median is the line segment drawn from a vertex to the midpoint of the opposite side. Understanding how to find midpoints and then equations of lines joining two points is essential in analytic geometry and appears frequently in competitive exams and school syllabi.

Given Data / Assumptions:

• Triangle ABC has vertices A(−5, 4), B(−4, 0), and C(−2, 2).
• D is the midpoint of side BC.
• AD is the median from vertex A to side BC.
• We need the equation of the line AD in standard form.

Concept / Approach:
First, we find the midpoint D of BC using the midpoint formula: D = ((x_B + x_C) / 2, (y_B + y_C) / 2). Then we find the equation of the line passing through A(−5, 4) and D using the two-point form or the slope-intercept method. Finally, we convert the result into a standard linear form and compare it with the given options to choose the correct one.

Step-by-Step Solution:
1) Coordinates of B are (−4, 0) and of C are (−2, 2). 2) Compute the midpoint D of BC: x_D = (−4 + (−2)) / 2 = −6 / 2 = −3, y_D = (0 + 2) / 2 = 2 / 2 = 1. So D is (−3, 1). 3) Now we need the line through A(−5, 4) and D(−3, 1). 4) Find the slope m: m = (1 − 4) / (−3 − (−5)) = (−3) / (2) = −3/2. 5) Use point-slope form with point A: y − 4 = (−3/2)(x + 5). 6) Multiply both sides by 2 to clear the fraction: 2(y − 4) = −3(x + 5). 7) Expand: 2y − 8 = −3x − 15. 8) Rearrange to standard form: 3x + 2y + 7 = 0, or 3x + 2y = −7.

Verification / Alternative check:
Check whether both points A and D satisfy the equation 3x + 2y = −7. For A(−5, 4), 3(−5) + 2(4) = −15 + 8 = −7, so A lies on the line. For D(−3, 1), 3(−3) + 2(1) = −9 + 2 = −7, so D also lies on the line. Therefore, 3x + 2y = −7 correctly represents the median AD, and our calculations are consistent.

Why Other Options Are Wrong:
Option a, 3x − 2y = −11, and option d, 3x − 2y = 11, use a different sign pattern and slope. Option b, 3x + 2y = 7, gives positive 7 on the right-hand side and does not pass through the given points. Substituting A or D into these incorrect equations will not satisfy them. Option e, 'None of these', is not correct because we have found a matching option. Only option c, 3x + 2y = −7, fits the median AD exactly.

Common Pitfalls:
Typical mistakes include miscalculating the midpoint, incorrectly computing the slope due to sign errors, or rearranging to standard form incorrectly. Some students also mix up the coordinates when using the point-slope formula. Writing each step clearly, especially the midpoint and slope, helps avoid these issues and leads to the correct median equation quickly.

Final Answer:
The equation of the median AD from A to side BC is 3x + 2y = -7.