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2012 AMC 10B Problem 3

2012 AMC 10B Problem 3AMC 10 Step by Step

Name: Date: September 13, 2026

Coordinate Geometry · difficulty 1 of 5 · about 1 min · position P1-10

The point in the xy-plane with coordinates $(1000, 2012)$ is reflected across the line $y = 2000$ . What are the coordinates of the reflected point?

A)
$(998,2012)$
B)
$(1000,1988)$
C)
$(1000,2024)$
D)
$(1000,4012)$
E)
$(1012,2012)$

Answer

Key insight

Reflecting over a horizontal line keeps x and sends y to 2(2000) - y, so 2012 goes to 1988.

Solution

The line $y = 2000$ is horizontal, so a reflection across it moves a point straight up or down: the $x$ -coordinate stays $1000$ .

The point sits $2012 - 2000 = 12$ units above the line, so its image sits $12$ units below it, at $y = 2000 - 12 = 1988$ .

The answer is $\boxed{\textbf{(B)}\ (1000, 1988)}$ .

Why this works

A reflection across $y = c$ fixes $x$ and replaces $y$ by $2c - y$ , because the line must be the midpoint of the point and its image. The same rule with roles swapped handles reflections across vertical lines $x = c$ .

The trap

Reflecting the x-coordinate instead, or moving the point to the mirror line itself instead of the same distance past it.

Common mistakes

Reflecting the x-coordinate instead, or moving the point to the mirror line itself instead of the same distance past it.
Adding the $12$ instead of subtracting it, giving $(1000, 2024)$ , choice (C).

Techniques

Set up the equation/formula and compute; no special trick needed