Line symmetry

1Summary 重點摘要

Fold in half, both sides match.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Part 1: What symmetry isWatch a butterfly: fold along a straight line right down the middle of its body, and the wings on the two sides cover each other exactly, shapes and patterns matching perfectly. That fold line is the line of symmetry, and the property of the two sides mirroring each other across it is called line symmetry. Why learn it? Symmetry is everywhere in nature and daily life: butterfly wings, human faces, many buildings and paper cuttings are symmetric, and knowing about symmetry helps us draw, cut and design. A common mistake is to think that two sides which look 'more or less alike' are symmetric; symmetry demands exact equality, and even a tiny difference means it is not symmetric.
2. Part 2: The fold testThe most direct way to test whether a figure has line symmetry is the fold test: fold the figure along a straight line, and if the two sides cover each other with nothing extra and nothing missing, that fold line is a line of symmetry. Why does folding work? Because symmetry means exactly that the two sides match across the line, and folding demonstrates this property with your own hands. For example, fold a cut-out heart vertically down the middle and the two halves cover each other, so the heart has a symmetry line; a figure that no fold can match onto itself is not symmetric. A common mistake is to try only one direction and then declare the figure asymmetric; try the main directions first. A parallelogram folds onto itself in no direction at all, which proves it has no line of symmetry.
3. Part 3: The square's four symmetry linesExperiment with a square sheet of paper: fold it once vertically and the halves match; fold it once horizontally and they match again; fold along each of the two diagonals and they still match. Four symmetry lines are found in all, so a square has 4 lines of symmetry. Why do the diagonal folds match? Because a square has four equal sides and four right angles; cutting along a diagonal gives two identical isosceles right triangles, equal in shape and size, so of course they cover each other. A common mistake is to think a square has only the vertical and horizontal lines and forget the two diagonals; another is to copy the conclusion from the rectangle and fold a general rectangle along a diagonal, which will not match, because a rectangle's diagonal is not a line of symmetry.
4. Part 4: How many lines common shapes haveThe numbers of symmetry lines for common shapes must be remembered clearly: a rectangle has only 2, one vertical and one horizontal; an equilateral triangle has 3, each passing through one corner and the midpoint of the opposite side; an isosceles triangle has only 1; the circle is the most special, since every straight line through its centre is a symmetry line, so it has infinitely many. Why does folding a rectangle along a diagonal fail? Because its length and width differ: for one 6 cm by 4 cm, the two triangles cut along the diagonal have equal areas but different side lengths, so their shapes are not the same and they cannot cover each other. A common mistake is to mix up rectangles and squares, thinking a rectangle also has 4; only the special square has 4, while a general rectangle has 2.
5. Part 5: Completing symmetric figures on grid paperThe way to complete a symmetric figure on grid paper is counting squares: for every key point of the pattern on the left of the axis, count how many squares it sits from the axis, then draw its partner point the same number of squares away on the right, and finally join the points in the same way. Why count square by square? Because two symmetric points must be equidistant from the axis and their joining line must cross it at a right angle; counting squares guarantees both conditions at once. For example, a point 3 squares from the axis gets its partner 3 squares away on the right, and a point 5 squares away gets a partner 5 squares away. A common mistake is to guess by feel, producing a figure of the wrong size; another is to move a point touching the axis one square outward, although a point at distance 0 stays exactly where it is.
6. Part 6: Symmetry around usSymmetry is everywhere in life: the letters A, H, O and X all have symmetry lines — A has one vertical line and H has two, vertical and horizontal; traffic signs, company logos, human faces, leaves and snowflakes are often symmetric too. Why do designers love symmetry? Because it gives a feeling of balance, stability and harmony; many logos arrange their patterns symmetrically and look calm and dignified. Learning to find symmetry lines not only helps judge figures but also helps us enjoy and create beauty: fold the paper before cutting, and unfolding reveals a symmetric paper cutting. A common mistake is to assume every letter has a symmetry line; S and G can never be folded onto themselves, so each figure must be tested in practice, never judged by impression alone.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Line symmetry: fold along the axis and the sides match
• Square 4 lines; rectangle 2
• Equilateral triangle 3; circle infinitely many
• A plain parallelogram has none
• Complete grid figures by counting squares from the axis