AMC 8 · 2010 · #6
Grade 4 geometry-2dProblem
Which of the following figures has the greatest number of lines of symmetry?
Pick an answer.
Toolkit + CCSS Solution
Understand
Restated: Among five named figures — an equilateral triangle, a non-square rhombus, a non-square rectangle, an isosceles trapezoid, and a square — which one has the most lines of symmetry?
Givens: Five candidate figures, one per answer choice; (A) equilateral triangle — three equal sides and angles; (B) non-square rhombus — four equal sides but unequal angles; (C) non-square rectangle — four right angles but adjacent sides unequal; (D) isosceles trapezoid — one pair of parallel sides plus equal legs; (E) square — four equal sides and four right angles
Unknowns: The figure (A)–(E) with the greatest number of lines of symmetry
Understand
Restated: Among five named figures — an equilateral triangle, a non-square rhombus, a non-square rectangle, an isosceles trapezoid, and a square — which one has the most lines of symmetry?
Givens: Five candidate figures, one per answer choice; (A) equilateral triangle — three equal sides and angles; (B) non-square rhombus — four equal sides but unequal angles; (C) non-square rectangle — four right angles but adjacent sides unequal; (D) isosceles trapezoid — one pair of parallel sides plus equal legs; (E) square — four equal sides and four right angles
Plan
Primary tool: #1 Draw a Diagram
Secondary: #2 Make a Systematic List
Counting lines of symmetry is a visual task, so Tool #1 (Draw a Diagram) is the natural primary tool: sketch each shape and try folding it along every candidate line — vertical, horizontal, and the two diagonals. Tool #2 (Make a Systematic List) keeps the bookkeeping clean: we list the five shapes and write down each shape's count, then read off the maximum. No algebra or coordinates are needed — this is a pure attribute-of-shapes question.
Execute — Answer: E
4.G.A.3 Step 1 - Sketch the equilateral triangle and try the fold lines.
- Each line from a vertex to the midpoint of the opposite side folds the triangle exactly onto itself.
- There are three vertices, so three such fold lines work.
💡 Folding a shape so it lands on itself is exactly the Grade 4 definition of a line of symmetry.
4.G.A.3 Step 2 - Sketch a non-square rhombus (a tilted diamond with all four sides equal, but angles not $90^\circ$).
- The two diagonals are fold lines and the figure lands on itself.
- But the vertical and horizontal lines through the center do NOT — the slanted sides do not match up.
- So the rhombus has exactly two lines of symmetry.
💡 Testing each candidate fold line one-by-one is still the Grade 4 symmetry standard.
4.G.A.3 Step 3 - Sketch a non-square rectangle (think of a door).
- The horizontal line through the middle and the vertical line through the middle both fold the rectangle onto itself.
- But the two diagonals do NOT — the long sides land on the short sides, which are different lengths.
- So the rectangle also has two lines of symmetry.
💡 Same Grade 4 fold-test — checking each candidate line directly on the drawing.
4.G.A.3 Step 4 - Sketch an isosceles trapezoid (one pair of parallel sides, equal legs).
- The only fold line that maps it onto itself is the vertical line through the midpoints of the two parallel sides.
- That gives one line of symmetry.
💡 Again the Grade 4 line-of-symmetry definition, applied to a specific quadrilateral.
4.G.A.3 Step 5 - Sketch a square.
- Both diagonals are fold lines (corner to corner), and both midlines (horizontal and vertical through the center) are fold lines.
- That gives four lines of symmetry in total.
💡 The square has both the rhombus's diagonals AND the rectangle's midlines as fold lines — a Grade 4 observation.
3.G.A.1 Step 6 - Make a systematic list and read off the maximum: triangle $3$, rhombus $2$, rectangle $2$, trapezoid $1$, square $4$.
- The largest count is $4$, belonging to the square.
💡 Listing each figure by its attributes (here, symmetry count) and comparing is the Grade 3 "shapes share attributes" idea.
4.G.A.3 Sketch the equilateral triangle and try the fold lines. Each line from a vertex 4.G.A.3 Sketch a non-square rhombus (a tilted diamond with all four sides equal, but ang 4.G.A.3 Sketch a non-square rectangle (think of a door). The horizontal line through the 4.G.A.3 Sketch an isosceles trapezoid (one pair of parallel sides, equal legs). The only 4.G.A.3 Sketch a square. Both diagonals are fold lines (corner to corner), and both midl 3.G.A.1 Make a systematic list and read off the maximum: triangle $3$, rhombus $2$, rect Review
Reasonableness: A regular polygon with $n$ sides has $n$ lines of symmetry. The equilateral triangle is regular with $n = 3$, the square is regular with $n = 4$ — so the square should beat the triangle by exactly one, matching our counts. The non-square rhombus and non-square rectangle each keep only half of the square's symmetries (diagonals OR midlines, not both), giving $2$ each. The isosceles trapezoid is the least symmetric of the four-sided figures and gets just $1$. Everything fits.
Alternative: Tool #5 (Find a Pattern): more "regularity" usually means more symmetry. Rank the shapes by how regular they look — square (most regular quadrilateral, $4$), equilateral triangle (most regular triangle, $3$), rhombus and rectangle (half-regular, $2$ each), isosceles trapezoid (least regular, $1$). The pattern points straight to (E).
CCSS standards used (min grade 4)
4.G.A.3Recognize a line of symmetry for a two-dimensional figure (Counting the lines of symmetry of each candidate figure by the fold-onto-itself test — the central skill of the whole problem.)3.G.A.1Understand that shapes in different categories share attributes (Comparing the five shape categories by a single shared attribute (number of lines of symmetry) and picking the largest.)
⭐ This AMC 8 problem only needs the Grade 4 "line of symmetry" idea you already know!
⭐ This AMC 8 problem only needs the Grade 4 "line of symmetry" idea you already know!