AMC 10 · 2012 · #3

Grade 5 geometry-2d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

reflection-symmetrycoordinate-geometry identify-subproblems ↑ Prerequisites: coordinate-geometry

📏 Medium solution 💡 2 insights

Problem

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?

Pick an answer.

(A)

(998,2012)

(B)

(1000,1988)

(C)

(1000,2024)

(D)

(1000,4012)

(E)

(1012,2012)

AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: A point sits at $(1000, 2012)$. Fold the plane along the horizontal line $y = 2000$. Find where the point lands after that fold.

Givens: The point is at $(1000, 2012)$.; The mirror line is $y = 2000$, a horizontal line.; Reflection means the mirror line is exactly halfway between a point and its image.

Unknowns: The coordinates of the reflected point.

Understand

Restated: A point sits at $(1000, 2012)$. Fold the plane along the horizontal line $y = 2000$. Find where the point lands after that fold.

Givens: The point is at $(1000, 2012)$.; The mirror line is $y = 2000$, a horizontal line.; Reflection means the mirror line is exactly halfway between a point and its image.

Plan

Primary tool: #1 Draw a Diagram

Secondary: #7 Identify Subproblems, #3 Eliminate Possibilities

Reflection is about position, so a quick sketch of the point and the horizontal mirror line makes the mirror image easy to see. Then split the work into two small questions: what happens to $x$, and what happens to $y$. A picture also lets you throw out choices that move in the wrong direction.

Execute — Answer: B

#1 Draw a Diagram 4.G.A.3 Step 1

Sketch the point and the mirror line

Draw the horizontal line $y = 2000$ and mark the point $(1000, 2012)$ above it.
A horizontal mirror flips things up and down, not left and right, so the left-right position ($x = 1000$) does not change.
Only the height ($y$) moves.

$$x = 1000 \text{ stays the same}$$

💡 Folding along a horizontal crease only swaps up for down, so the sideways position is untouched.

#7 Identify Subproblems 4.NBT.B.4 Step 2

Measure the gap to the line

Find how far the point is above the mirror line by subtracting the two heights.
The point is at height $2012$ and the line is at height $2000$, so the gap is $2012 - 2000 = 12$.

$$2012 - 2000 = 12$$

💡 The distance from a point to a horizontal line is just the difference of their heights.

#7 Identify Subproblems 4.NBT.B.4 Step 3

Drop the same distance below

A reflection keeps the same distance on the other side of the line.
The point was $12$ above the line, so its image is $12$ below the line.
Start at the line's height $2000$ and go down $12$: $2000 - 12 = 1988$.

$$2000 - 12 = 1988$$

💡 The mirror line sits exactly halfway, so the image is the same gap on the far side.

#3 Eliminate Possibilities 5.G.A.1 Step 4

Write the reflected coordinates

Put the unchanged $x$ together with the new $y$.
The reflected point is $(1000, 1988)$.
Choices that change $x$ or move the point up can be crossed out, since a horizontal mirror only moves the point straight down here.
This matches answer (B).

$$(1000,\ 1988)$$

💡 An ordered pair pins the point down: keep the steady coordinate, swap in the mirrored one.

[1] #1 4.G.A.3 Draw the horizontal line $y = 2000$ and mark the point $(1000, 2012)$ above it.

[2] #7 4.NBT.B.4 Find how far the point is above the mirror line by subtracting the two heights.

[3] #7 4.NBT.B.4 A reflection keeps the same distance on the other side of the line. The point wa

[4] #3 5.G.A.1 Put the unchanged $x$ together with the new $y$. The reflected point is $(1000,

Review

Reasonableness: The line $y = 2000$ should be the midpoint of the point and its image in height. The two heights are $2012$ and $1988$, and their average is $(2012 + 1988) / 2 = 2000$, exactly the mirror line. The $x$-coordinate stayed $1000$, as a horizontal mirror requires. So $(1000, 1988)$, answer (B), is consistent.

Alternative: Use the midpoint idea directly: the mirror height $2000$ must be halfway between the old height $2012$ and the new height $h$, so $2000 = (2012 + h)/2$, giving $h = 4000 - 2012 = 1988$. Same result without measuring the gap first.

CCSS standards used (min grade 5)

4.G.A.3 Recognize a line of symmetry for a two-dimensional figure (Seeing that a horizontal mirror line keeps the left-right position and flips only the height.)
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers (Computing the distance $2012 - 2000 = 12$ and the new height $2000 - 12 = 1988$.)
5.G.A.1 Use a pair of perpendicular number lines forming a coordinate system (Reading and writing the point as an ordered pair $(x, y)$ in the coordinate plane.)

⭐ To mirror a point across a flat line, keep the sideways number and put the height the same distance on the other side.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Sketch the point and the mirror line

Measure the gap to the line

Drop the same distance below

Write the reflected coordinates

Review

CCSS standards used (min grade 5)

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