AMC 10 · 2019 · #3

Grade 8 geometry-2d

Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In

rotation-isometrycoordinate-geometryreflection-symmetry systematic-enumerationpattern-recognition ↑ Prerequisites: coordinate-geometryrotation-isometry

📏 Short solution 💡 2 insights

Problem

A segment has endpoints at (-2,1) and (-1,4). A rigid transformation moves it so the first lands at (2,-1) and the second at (1,-4). Decide which of the five listed transformations does exactly that.

Pick an answer.

(A)

reflection in the } y \text{-axis

(B)

counterclockwise rotation around the origin by } 90^{\circ

(C)

translation by } 3 \text{ units to the right and } 5 \text{ units down

(D)

reflection in the } x \text{-axis

(E)

clockwise rotation about the origin by } 180^{\circ

How to solve

Strategy Eliminate Possibilities

There are only five candidates and each one is a single coordinate rule, so Tool #3 (Eliminate Possibilities) is the efficient frame: convert each choice into a rule, apply it to one point, and discard whatever misses. Before testing, Tool #5 (Look for a Pattern) compares A with A' and B with B' to read the actual rule off the data, which turns the search into a matching job. Tool #1 (Draw a Diagram) and Tool #17 (Visualize Spatial Relationships) back that up: plotting the four points shows the segment landing on the far side of the origin, which is the picture a half-turn makes.

1STEP 1

Line up each point with its image

Line each point up with its image side by side.

A(-2,1)→ A'(2,-1), B(-1,4)→ B'(1,-4)

2STEP 2

Read the coordinate rule

Both coordinates simply flip sign.

(x,y)↦(-x,-y)

3STEP 3

Turn each choice into a rule

Turn each choice into a rule.

(A) (-x,y) (B) (-y,x) (C) (x+3,y-5) (D) (x,-y) (E) (-x,-y)

4STEP 4

Test the choices on A

One point filters out most of them.

(A)→(2,1), (B)→(-1,-2), (C)→(1,-4), (D)→(-2,-1)

5STEP 5

Watch the translation trap

The translation choice swaps the points.

(C): A↦(1,-4)=B', B↦(2,-1)=A'

6STEP 6

Confirm the survivor

What survives is a half turn about the origin.

(-2,1)↦(2,-1)=A', (-1,4)↦(1,-4)=B' → (E)

Answer

clockwise rotation about the origin by 180°

The answer survives three independent checks. First, both endpoints obey (x,y)↦(-x,-y), and the correspondence is the one the problem asked for. Second, the motion is genuinely rigid: AB=√((-1-(-2))²+(4-1)²)=√(10) and A'B'=√((1-2)²+(-4-(-1))²)=√(10), so lengths are preserved, as an isometry requires. Third, the midpoint of AA' is ((-2+2)/2,(1+(-1))/2)=(0,0) and the midpoint of BB' is ((-1+1)/2,(4+(-4))/2)=(0,0) — every point and its image straddle the origin, which is precisely the signature of a half-turn about the origin.

💡Key takeaway

Compare each point with its image first — when both coordinates just flip sign, the segment has been spun a half-turn around the origin.

Line up each point with its image
Read the coordinate rule
Turn each choice into a rule
Test the choices on A
Watch the translation trap
Confirm the survivor