Point P is the midpoint of the line segment AB. If P has coordinates (5, -1) and point A has coordinates (2, -4), find the coordinates of point B. Use the midpoint formula and give the exact ordered pair for B.

Introduction / Context: This problem tests the midpoint formula in coordinate geometry. If a point P is the midpoint of segment AB, then P lies exactly halfway between A and B. That means P’s x-coordinate is the average of the x-coordinates of A and B, and similarly for the y-coordinates. Many aptitude problems reverse the usual direction: instead of finding the midpoint, you are given the midpoint and one endpoint and must find the other endpoint.

Given Data / Assumptions:

• Midpoint P = (5, -1) • Endpoint A = (2, -4) • Endpoint B = (xB, yB) to be found

Concept / Approach: Use the midpoint formula: Midpoint P = ((xA + xB)/2, (yA + yB)/2). Solve two simple equations: (xA + xB)/2 = xP and (yA + yB)/2 = yP. Then isolate xB and yB by multiplying by 2 and subtracting the known endpoint coordinates.

Step-by-Step Solution: 1) Write the midpoint equations: (2 + xB)/2 = 5 (-4 + yB)/2 = -1 2) Solve for xB: 2 + xB = 10 xB = 10 - 2 = 8 3) Solve for yB: -4 + yB = -2 yB = -2 + 4 = 2 4) Therefore, point B is: (8, 2)

Verification / Alternative check: Check by recomputing midpoint of A(2, -4) and B(8, 2): ((2 + 8)/2, (-4 + 2)/2) = (10/2, -2/2) = (5, -1), which matches P exactly. So the coordinates are correct.

Why Other Options Are Wrong: • Any other option fails the midpoint check and would not average with A to give (5, -1).

Common Pitfalls: • Forgetting to multiply by 2 before subtracting. • Mixing up x and y while solving the two equations.

Final Answer: (8, 2)