AMC 10 · 2015 · #8

Grade 8 arithmetic
Archetype Paper Fold-and-Cut Symmetry └ Sub-archetype Composition of Reflections of a Pictured Figure
rotation-isometryreflection-symmetrytransformations-compositionspatial-visualization identify-subproblems ↑ Prerequisites: rotation-isometryreflection-symmetry
📏 Medium solution 💡 2 insights 📊 Diagram
Problem
A letter F sits in the plane. It is put through three moves in order: a 90° clockwise rotation about the origin, then a reflection across the y-axis, then a half turn (180°) about the origin. Decide which of the five pictured F's shows where it lands.

Pick an answer.

(A)
Image choice (A) — see answer-choice figure
(B)
Image choice (B) — see answer-choice figure
(C)
Image choice (C) — see answer-choice figure
(D)
Image choice (D) — see answer-choice figure
(E)
Image choice (E) — see answer-choice figure

AMC 10 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Visualize Spatial Relationships

You cannot picture three turns-and-flips stacked on top of each other all at once, so Tool #17 (Visualize Spatial Relationships) is the engine: do the moves one at a time and watch the F's pose change. To keep the picture honest, Tool #1 (Draw a Diagram) pins the F down with just two arrows — the spine (which way the long stroke points) and the arm (which way the teeth point) — because those two arrows, plus knowing the F can be flipped to a mirror image, completely fix its orientation. Tracking each move with a coordinate rule like (x,y)↦(y,-x) keeps the mental rotation from drifting. After the third move, Tool #3 (Eliminate Possibilities) matches the final pose — and crucially its mirror-image 'handedness' — against the five pictures to pick the one that fits.

1STEP 1

Pin the F down with two arrows

Describe the starting F by two arrows — its spine points up, arms point right — enough to fix how a not-yet-mirrored F is turned.

spine → +y, arms → +x
2STEP 2

Move 1: rotate 90 degrees clockwise

A 90° clockwise turn sends up to right and right to down, so the spine now points right and the arms point down.

(x,y)↦(y,-x): spine→ +x, arms→ -y
3STEP 3

Move 2: reflect across the y-axis

Reflecting across the y-axis reverses left-right only: the spine flips to left, arms stay down — and the F is now a backwards mirror image.

(x,y)↦(-x,y): spine→ -x, arms→ -y
4STEP 4

Move 3: half turn about the origin

A half turn reverses both directions, so the spine swings back to right and the arms point up.

(x,y)↦(-x,-y): spine→ +x, arms→ +y
5STEP 5

Read the final F and match it

The three moves compose to a single mirror flip across y=x — spine right, arms up — and only choice (E) shows that flipped, lying-down F.

net move=(x,y)↦(y,x); spine→ +x, arms→ +y → (E)
Answer
Image choice (E) — see answer-choice figure
Combine the three moves into one to confirm. Multiplying the matrices in order (rotate 90° CW, then reflect in y, then 180°) collapses to (x,y)↦(y,x), the reflection across y=x. Its determinant is -1, so the result is a single mirror flip — matching the fact that we used one reflection and two rotations (one orientation-reversing move total). Reflecting the upright F across y=x does send its upward spine to the right and its rightward arms upward, exactly the pose of choice (E). Both the step-by-step arrow tracking and the one-shot matrix agree, so (E) is solid.
💡Key takeaway

Don't juggle all three moves at once — follow just two arrows, the spine and the teeth, one move at a time, and remember a single flip leaves the F mirror-backwards.

  • Pin the F down with two arrows
  • Move 1: rotate 90 degrees clockwise
  • Move 2: reflect across the y-axis
  • Move 3: half turn about the origin
  • Read the final F and match it

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