A shape (let us consider only two dimensional shapes in this topic) is said to be symmetrical if any line or any point can reflect the portion on one side exactly same as the portion on the other side. If the symmetry is around a line, then that line is called line of symmetry and if the symmetry is around a point, then that line is called point of symmetry. A shape which has a point of symmetry is said to have a point symmetry.

If a shape has a point symmetry, then any point on the shape has a matching point which is exactly at the same distance from the point of symmetry but in the opposite direction. The glaring example of a shape that has a point of symmetry is a circle.

Similarly many two dimensional shapes can be cited as examples. But they have a greater application in real life situations. For example, a rotating object can have a minimal load on the point of rotation (in practice, the bearings) and keep it safe only if the point of rotation is same as the point symmetry of that shape.

Let us study the fundamentals of point symmetry in subsequent sections.

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The above diagram shows some shapes which have point symmetry shown by the red dot. In each case, (except in fig (i), which is obvious), it is shown that any point on the shape has a matching point which is exactly at the same distance from the point of symmetry but in the opposite direction.

Graphs of all odd functions have point symmetry. The point of symmetry need not be at origin and may be else where as shown in figure (iv).

The point of symmetry need not necessarily be on the shape itself, like we see in fig (ii).

Another important point is the shapes with point symmetry also have a rotational symmetry of order 2. That is, the shape remains exactly the same when it is rotated by 180^{o} over the point of symmetry.

Example 1: Which of the following shapes have point symmetry?

The shape in figure (i) has a symmetry over x-axis. But when the shape is rotated by 180^{o} around point O, it becomes a different shape. Hence this shape does not have point symmetry.

The shape in figure (ii) has a symmetry neither over x-axis nor over y-axis. But when the shape is rotated by 180^{o} around point O, the shape remains the same. Also it can be seen that any point on the shape has a matching point on the opposite direction at the same distance. Hence this shape has a point symmetry at O.

The shape in figure (iii) has a symmetry over y-axis. But when the shape is rotated by 180^{o} around point O, it becomes a different shape. Hence this shape does not have point symmetry.

Example 2: Which of the following shapes have point symmetry?

The number 8 has a symmetry over x-axis and also over y-axis. Also when the shape is rotated by 180^{o} around point origin, the shape remains the same. Also it can be seen that any point on the shape has a matching point on the opposite direction at the same distance. Hence this shape has a point symmetry at origin.

The letter A has a symmetry over y-axis. But when the shape is rotated by 180^{o} around origin, it becomes a different shape. Hence this shape does not have point symmetry.

The letter B has a symmetry over x-axis. But when the shape is rotated by 180^{o} around origin, it becomes a different shape. Hence this shape does not have point symmetry.

The letter Z has a symmetry neither over x-axis nor over y-axis. But when the shape is rotated by 180^{o} around point origin, the shape remains the same. Also it can be seen that any point on the shape has a matching point on the opposite direction at the same distance. Hence this shape has a point symmetry at origin.