Point P is the midpoint of segment AB. The coordinates of P are (3, 1) and the coordinates of B are (5, -4). What are the coordinates of point A?

Introduction / Context:
The midpoint formula is a fundamental concept in coordinate geometry. It gives the coordinates of the point exactly halfway between two given points. This question reverses that idea: you are given the midpoint and one endpoint and must find the other endpoint. Understanding how to manipulate the midpoint formula is essential for solving such problems efficiently.

Given Data / Assumptions:

• Segment AB has midpoint P.
• Coordinates of P are (3, 1).
• Coordinates of B are (5, -4).
• We must determine the coordinates of A(x, y).

Concept / Approach:
If A(x1, y1) and B(x2, y2) are endpoints of a segment, the midpoint P has coordinates:
P = ( (x1 + x2) / 2 , (y1 + y2) / 2 ) Here P and B are known, and we need A. So we can set up two equations by equating the x and y coordinates of the midpoint to the given values and then solve for the unknown coordinates of A.

Step-by-Step Solution:
Step 1: Let A have coordinates (x, y). Step 2: Use the midpoint formula for the x coordinate: (x + 5) / 2 = 3. Step 3: Multiply both sides by 2: x + 5 = 6, so x = 6 - 5 = 1. Step 4: Use the midpoint formula for the y coordinate: (y + (-4)) / 2 = 1. Step 5: Simplify the numerator: (y - 4) / 2 = 1. Step 6: Multiply both sides by 2: y - 4 = 2, so y = 2 + 4 = 6. Step 7: Therefore, A has coordinates (1, 6).

Verification / Alternative check:
Check that P really is the midpoint of A(1, 6) and B(5, -4). The average of the x coordinates is (1 + 5) / 2 = 6 / 2 = 3. The average of the y coordinates is (6 + (-4)) / 2 = 2 / 2 = 1. This matches the given midpoint (3, 1), so our solution is consistent and correct.

Why Other Options Are Wrong:

• (-1, 7), (1, -7), and (-1, -7): These do not produce a midpoint of (3, 1) when paired with B(5, -4).
• (7, -1): This would give a midpoint with x coordinate (7 + 5) / 2 = 6, not 3.

Common Pitfalls:
Some learners mistakenly subtract instead of averaging or mis-handle negative coordinates. Another issue is mixing x and y equations when solving for the unknown point. Always apply the midpoint formula component wise: solve separately for the x coordinate and the y coordinate using the average method.

Final Answer:
Thus, the coordinates of point A are (1, 6).