Reflecting points

So far we have said that when a figure is folded along a line of symmetry, the two parts overlap. There is another way to say this: the part on one side of the line is reflected by the line to the other side, just like in a mirror, and the part on the other side is reflected back to the first side. Cover half a picture of a butterfly and stand a small mirror on the line: the mirror shows the missing half, and the whole butterfly is back.

To see this, the book labels the corners of a square. Picture the square with A at the top left, B at the top right, C at the bottom right and D at the bottom left. Reflect it in the vertical line of symmetry. B and C, on the right, are reflected to the left side and take the places where A and D were. A goes to B's place and D goes to C's place.

Now reflect in the diagonal from A to C. A and C lie on the line, so they stay where they are. B and D are the same distance from the line on opposite sides, so they swap: B goes to D's place and D to B's place. Reflect in the horizontal line instead, and the top swaps with the bottom: A goes to D's place, D to A's, B to C's and C to B's.

A figure that has a line or lines of symmetry is also said to have reflection symmetry.