Symmetry

The axis or line of symmetry is an imaginary line that runs through the centre of a shape creating two perfectly identical halves.

The other hole is as shown in figure.

The lines of symmetry are marked in the figure:

To make the figure symmetrical about the diagonals, we need to shade the figure in the following way:

Yes, the figure will be symmetrical about both the diagonals.

Yes, the figure can be made symmetrical by more than one way which is shown as follows:

(a) To complete the figure, we need to join the squares exactly the same way as the shape is in on the other side of the line of symmetry. Hence, the completed figure is shown as follows:

(b) To complete the figure, we need to join the squares exactly the same way as the shape is in on the other side of the line of symmetry. Hence, the completed figure is shown as follows:

(c) To complete the figure, we need to join the squares exactly the same way as the shape is in on the other side of the line of symmetry. Hence, the completed figure is shown as follows:

(d) To complete the figure, we need to join the dots exactly the same way as the shape is in on the other side of the line of symmetry. Hence, the completed figure is shown as follows:

a) Letters of the English alphabet symmetric about a vertical mirror are: -

A, H, I, M, O, U, V, W, X and Y

These letters when flipped horizontally along a mirror retains the original figure which is shown as follows:

b) Letters of the English alphabet symmetric about a horizontal mirror: -

B, C, D, E, H, I, O and X

The letters when flipped vertically along a mirror retains the original figure which is shown as follows:

c) Letters of the English alphabet symmetric about both horizontal and vertical mirrors: -

H, I, O and X

The letters when flipped along both the axes (horizontal and vertical) retains the original figure.

The shapes which cannot be divided into two identical halves have no line of symmetry.

Three examples with no line of symmetry are:

a) A scalene triangle

b) A quadrilateral

c) An irregular polygon

The line of symmetry of:

a) an isosceles triangle – Median

The median divides a triangle into two equal halves.

b) a circle – Diameter

The diameter divides a circle into two equal semi-circles.

Figure (a) has 4 rotational symmetry.


Figure (b) has 3 rotational symmetry.



Figure (d) has 2 rotational symmetry



Figure (e) has 3 rotational symmetry.



Figure (f) has 4 rotational symmetry.
The figures having rotational symmetry are (a), (b), (d), (e) and (f).

The figures when rotated completely more than once, looks exactly the same as the original figure. Also, all these figures have more than one line of symmetry. Figures having multiple lines of symmetry also have rotational symmetry more than one.

A square and a circle have both line symmetry and rotational symmetry.

a) An equilateral triangle has both line and rotational symmetries of order more than 1.

Line Symmetry:

Rotational Symmetry:

b) An isosceles triangle has only one line of symmetry and rotational symmetry of order 1.

Line Symmetry:

Rotational Symmetry:

c) Not possible. A quadrilateral with a line symmetry may have rotational symmetry of order one but not more than one.

d) A trapezium has a line symmetry and a rotational symmetry of order one.

Line Symmetry:

Rotational Symmetry:

Yes, a figure having two or more lines of symmetry will have rotational symmetry of order more than one since, both the line symmetry and the rotational symmetry are symmetric about the centre axis.

A square has a line and rotational symmetry of order more than one.

Line Symmetry:

Rotational Symmetry:

The other angles will be 120°, 180°, 240°, 300°, 360°

Since, the figure is said to have rotational symmetry about same angle as the first one. Hence, the figure will look exactly the same when rotated by 60° from the last position.

i. If the angle of rotation is 45°, then it is possible to have rotational symmetry of order more than one since, 360° is completely divisible by 45°.

ii. If the angle of rotation is 17°, then it is not possible to have rotational symmetry of order more than one since, 360° is not completely divisible by 17°.