Problem 6 · 2010 AMC 8
Easy
Geometry & Measurement
lines-of-symmetry
Which of the following figures has the greatest number of lines of symmetry?
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Answer: E — Square (4 lines).
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Hint 1 of 2
A line of symmetry is a fold line: fold the shape along it and the two halves land exactly on each other. Picture the fold for each figure instead of trying to remember a rule.
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Hint 2 of 2
The more ‘sameness’ a shape has (equal sides, equal angles), the more fold lines work. So check the most regular shape — the square — first.
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Approach: count fold lines, most-regular first
- Fold-test the square first since it looks most regular: it matches across both diagonals and both midpoint lines — 4 lines.
- The others fall short: equilateral triangle 3, non-square rhombus 2 (only its diagonals), non-square rectangle 2 (only its midpoint lines), isosceles trapezoid just 1.
- Most lines: square.
- Worth keeping: a regular n-sided shape has exactly n lines of symmetry (triangle 3, square 4, pentagon 5…). Knowing that, you can often spot the winner without drawing a single fold.
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