AMC 10 · 2012 · #3
Grade 5 geometry-2dPick an answer.
AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Sketch the point and the mirror line
Draw the line y = 2000 and mark (1000, 2012) above it — a horizontal fold flips up and down only, so x = 1000 stays.
Folding along a horizontal crease only swaps up for down, so the sideways position is untouched.
4.G.A.3Draw A DiagramMeasure the gap to the line
Subtract the two heights: the point sits at 2012 and the line at 2000, so the gap is 12.
The distance from a point to a horizontal line is just the difference of their heights.
4.NBT.B.4Identify SubproblemsDrop the same distance below
The image keeps the same gap on the far side, so drop 12 from 2000: the new height is 1988.
The mirror line sits exactly halfway, so the image is the same gap on the far side.
The mirror line sits exactly halfway, so the image is the same gap on the far side.
▸ Why?
A fold line meets what it folds at right angles and cuts the gap exactly in half.
▸ Why?
Folding moves the point without stretching anything, so the two gaps are the same length.
Write the reflected coordinates
Pair the unchanged x with the new y: (1000, 1988). Choices that shift x or move the point up are out — answer (B).
An ordered pair pins the point down: keep the steady coordinate, swap in the mirrored one.
5.G.A.1Eliminate PossibilitiesTo mirror a point across a flat line, keep the sideways number and put the height the same distance on the other side.
- Sketch the point and the mirror line
- Measure the gap to the line
- Drop the same distance below
- Write the reflected coordinates
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