Line symmetry

A shape has line symmetry if you can fold it along some line and the two halves match up exactly. The fold line is called the line of symmetry (or axis of symmetry, or mirror line).

Concept

Imagine a butterfly. Draw a vertical line down its middle. The left wing is a mirror image of the right wing. If you folded the picture along that line, the left wing would land exactly on the right wing. That line is the butterfly's line of symmetry.

A more precise definition: a line $\ell$ is a line of symmetry of a shape $S$ if for every point $P$ in $S$, the mirror image of $P$ across $\ell$ is also in $S$. The shape and its mirror image are identical.

Some shapes have one line of symmetry, like:

• A letter "T" (vertical line down the middle).
• A trapezium (only if it is isosceles , the perpendicular bisector of the parallel sides).
• The number "3" (horizontal line).
• A heart shape (vertical down the middle).

Some shapes have multiple lines of symmetry (we will see those in the next subtopic). And some shapes have none , like the letter "F" or "P".

Finding the line of symmetry

Three reliable ways:

1. Fold. Cut the shape out of paper and try folding. If the two halves match, you've found a line of symmetry.
2. Mirror. Place a small mirror perpendicular to the page along a candidate line. If the half-image in the mirror combined with the half on the page recreates the original figure, that is a line of symmetry.
3. Symmetry of points. For every important point on the shape (corners, special marks), check whether its mirror image (across the candidate line) is also a point on the shape.

Examples of line-symmetric shapes

• Equilateral triangle. $3$ lines of symmetry (from each vertex to the midpoint of the opposite side).
• Square. $4$ lines (two diagonals plus two midline axes).
• Rectangle (not a square). $2$ lines (the two midline axes , not the diagonals).
• Circle. Infinitely many lines, all passing through the centre.
• Regular pentagon. $5$ lines.
• Rhombus (not a square). $2$ lines (the two diagonals).

Worked examples

Example 1. Does the letter "A" have a line of symmetry? If so, sketch it.

• Yes, a vertical line down the middle. Folding along it makes the two halves match.

Example 2. Does the letter "F" have a line of symmetry?

• No. Try every direction , no fold makes the halves match.

Example 3. How many lines of symmetry does an equilateral triangle have? Sketch them.

• $3$ lines. Each line passes from a vertex perpendicular to the opposite side.

Example 4. Does a parallelogram (which is not a rectangle or rhombus) have any line of symmetry?

• No. None of its sides or diagonals work as a fold line. (A general parallelogram has rotational symmetry but no line symmetry.)

Example 5. Find lines of symmetry of the number "8".

• $2$ lines: one horizontal (through the middle), one vertical.

Try it yourself

1. Sketch a butterfly with one clear line of symmetry. Where is it?
2. Does the letter "H" have line symmetry? How many lines?
3. List the capital letters of the alphabet that have at least one line of symmetry.
4. Sketch an isosceles triangle and draw its line of symmetry.
5. How many lines of symmetry does a regular hexagon have?
6. Does a kite shape have a line of symmetry? Sketch one.
7. Look at the front view of any car. Does it usually have a line of symmetry? Where?
8. Investigate: sketch the digit "6". Does it have line symmetry? What about "0"?

Activity

Inkblot fold. Take a small sheet of paper and fold it in half. Open it. Drop a few drops of paint or ink on one side of the fold. While the ink is wet, refold and press gently. Open the paper. You will get a beautiful symmetric design , by definition, both halves are mirror images across the fold line. Try it with different colours.