In parallelogram ABCD with vertices A(5, 0), B(−2, 3) and C(−1, 4), determine the equation of side AD in coordinate geometry.

Introduction / Context:
This problem tests your understanding of coordinate geometry, specifically properties of a parallelogram and how to find the equation of a line when you know two points that lie on it. Such questions combine vector or midpoint ideas with straight line equations and are very popular in analytical geometry sections of exams.

Given Data / Assumptions:

• Parallelogram ABCD has vertices A(5, 0), B(−2, 3) and C(−1, 4).
• We must find the equation of side AD.
• Vertices follow the order A → B → C → D around the parallelogram.

Concept / Approach:
Key concepts:

• In a parallelogram, the diagonals bisect each other, and opposite sides are parallel.
• For a parallelogram, position vectors satisfy A + C = B + D, so D = A + C − B.
• Once two points on line AD are known, its equation can be found using the two point or point slope form.

Step-by-Step Solution:
Use the vector relation: D = A + C − B. Compute A + C = (5, 0) + (−1, 4) = (4, 4). Subtract B: D = (4, 4) − (−2, 3) = (4 + 2, 4 − 3) = (6, 1). So vertex D is at (6, 1). Line AD passes through A(5, 0) and D(6, 1). Slope m = (1 − 0) / (6 − 5) = 1. Using point slope form with point A: y − 0 = 1(x − 5). Therefore y = x − 5 is the equation of side AD.

Verification / Alternative check:
Check that C also lies on a line parallel to AD. Slope of BC is (4 − 3) / (−1 − (−2)) = 1 / 1 = 1, the same as slope of AD. Since opposite sides of a parallelogram are parallel, this supports the correctness of side AD having slope 1 and equation y = x − 5 through point A.

Why Other Options Are Wrong:
Option a: y = 2x − 5 has slope 2, not matching the slope between A and D.
Option b: y = x + 5 passes through (0, 5), not through point A(5, 0).
Option c: y = 2x + 5 has both wrong slope and wrong intercept.
Option e: y = −x + 5 has negative slope and does not pass through A(5, 0) or D(6, 1).

Common Pitfalls:
A common mistake is to misidentify the coordinates of D by adding instead of subtracting correctly, or to assume that C is diagonally opposite A without verifying the order of vertices. Another frequent error is mixing up x and y differences when computing slope. Carefully applying the vector relation and slope formula avoids these issues.

Final Answer:
The equation of side AD of the parallelogram is y = x − 5.