Difficulty: Easy
Correct Answer: (1, 6)
Explanation:
Introduction:
This question uses the midpoint formula from coordinate geometry to find the unknown endpoint of a line segment when one endpoint and the midpoint are known. Mastering this formula is essential for many geometry and analytic geometry problems.
Given Data / Assumptions:
Concept / Approach:
If A has coordinates (x₁, y₁) and B has coordinates (x₂, y₂), then the midpoint M of AB has coordinates: M = ( (x₁ + x₂) / 2 , (y₁ + y₂) / 2 ). We know M and B, so we can set up equations and solve for x₁ and y₁, the coordinates of A.
Step-by-Step Solution:
Let A = (x, y) and B = (5, −4). Midpoint P = (3, 1) is given by ((x + 5) / 2, (y − 4) / 2). Equate x coordinate of midpoint: (x + 5) / 2 = 3. Multiply both sides by 2: x + 5 = 6, so x = 1. Equate y coordinate of midpoint: (y − 4) / 2 = 1. Multiply both sides by 2: y − 4 = 2, so y = 6. Therefore, A = (1, 6).
Verification / Alternative check:
Check that the midpoint of A(1, 6) and B(5, −4) is indeed (3, 1). Average of x coordinates: (1 + 5) / 2 = 6 / 2 = 3. Average of y coordinates: (6 + (−4)) / 2 = 2 / 2 = 1. This matches P exactly, so the answer is correct.
Why Other Options Are Wrong:
Each alternative pair of coordinates, such as (−1, 7) or (1, −7), fails either the x coordinate or y coordinate midpoint condition, or both. Only (1, 6) produces midpoint (3, 1) when paired with B(5, −4).
Common Pitfalls:
A typical mistake is to subtract instead of adding coordinates, or to forget to divide by 2. Another error is mixing up which point is A and which is B, though the midpoint formula is symmetric so this usually does not change the result. Writing the formula explicitly helps prevent errors.
Final Answer:
The coordinates of point A are (1, 6).
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