What is a line of symmetry?

Mirror halves

Draw a triangle with two equal sides and fold it along the line through its top corner and the middle of its base. One half covers the other completely. The two parts are called mirror halves, and the fold line is a line of symmetry. Now take four jigsaw pieces fitted together and fold along a line through the middle: the left half does not fit over the right half, so that line is not a line of symmetry.

A line that cuts a figure into two parts that exactly overlap when folded along that line is a line of symmetry, or axis of symmetry, of the figure.

How many lines of symmetry?

Take a square piece of paper and fold it in half vertically, then horizontally, then along one diagonal and then along the other, opening it out each time. Each fold makes the halves overlap, and there is no other way to do it. So a square has 4 lines of symmetry.

Figure	Lines of symmetry	Where they are
Isosceles triangle (two equal sides)	1	From the top corner to the middle of the base
Rectangle (not a square)	2	Through the middles of opposite sides
Equilateral triangle	3	From each corner to the middle of the opposite side
Square	4	Through the middles of opposite sides, and both diagonals
Regular hexagon	6	Through opposite corners, and through middles of opposite sides
Scalene triangle	0	None

Notice the rectangle. Fold it along a diagonal and the corners do not meet, so a rectangle’s diagonals are not lines of symmetry, although a square’s diagonals are.

Reflection

A line of symmetry acts like a mirror: it reflects the part of the figure on one side onto the other side. Label the corners of a square A, B, C and D, going round from the top left. Reflect it in the vertical line of symmetry: B and C move to the places of A and D, and A and D move to the places of B and C. Reflect it in the diagonal AC: A and C stay put, while B and D swap. A figure that has a line of symmetry is said to have reflection symmetry.

Finding a mirror image

The mirror image of a point is on the other side of the line of symmetry, at the same distance from it, on a line at right angles to it. To complete a figure on squared paper, copy each corner across the line by counting squares, then join the new corners in the same order. For a slanting line, turn the paper so the line looks upright, or count squares along the slant.

Making symmetric shapes

• Ink blots: drop some ink on one half of a folded paper, press the halves together and open it. The blot is symmetric about the fold.
• Folding and cutting: cut a shape from a folded sheet and unfold it. The fold becomes a line of symmetry of the hole.
• Punching game: punch a hole in a folded square. Unfolded, the two holes are mirror images across the fold. Fold twice before punching and one punch gives four holes.

Lines of symmetry around us

Look for lines of symmetry in leaves, butterflies, kolams, rangolis, letters such as A, H and M, and the front views of buildings like the Taj Mahal. A cloud or a scalene triangle has none. Some figures, like a pinwheel, have no line of symmetry but still look the same when turned; learn about them in rotational symmetry.