Definition Of Rotational Symmetry

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An image is said to have rotational symmetry if it gives the same image after rotating the original image about a point by some angle.

Here when the image is rotated about the centre point, after rotating it by 90° we can find the same image as earlier. In this way we can find the image at 180°,270° and 360° also. So totally the image repeats 4 times in the range 0° to 360°. The image shows 4th order rotational symmetry

Generally the image may get repeated one or more times at different angles. The number of times the image repeated is known as the order of Rotational symmetry. And the point about which the image is rotated is known as vertex.

Note: The angle of rotation must be in the range of 0°-360°degrees.Because the same counted image again gets repeated after 360°.So it gives repeated count.
Explanation of Rotational Symmetry:

The angles of symmetry can be calculated easily by knowing the order of symmetry. The following formula helps in finding the angles.

An = `360^0/n` , n = order & An = Angle of symmetry.

Now multiply the obtained value starting from 1 till ...
... the order is reached. Each gives the angle of symmetry.

If the figure gets repeated only once after 360°, then it can’t be said as order 1 rotational symmetry. Because every figure repeats itself after 360° of rotation. So order 1 symmetry doesn’t exist.

Problem Based on Rotational Symmetry :

Now lets try to solve some model problems on Rotational Symmetry.

Ex 1) Find the order of symmetry and angles of the following figure.

Problem figure for Rotational Symmetry

Sol: Here we can observe that the figure gives the same replica 3 times when rotated about the vertex. So order = 3.

Angles of symmetry can be obtained by using the above mentioned formula.

i.e., An = `360^0/n` .

Put n = 3(order) `rArr` An = `360^0/3` = 120° is the initial angle of symmetry.

Now multiply obtained An value with 1, 2, 3.We get

120*1 = 120°

120°*2 = 240°

120°*3 = 360°.Here the order is reached. So stop putting the values. Now we have obtained 3 angles of symmetry.

Check this Binomial Distribution Mean awesome i recently used to see.

Ex :2) Similarly for the figure find the rotataional symmetry:

Problem figure for Rotational Symmetry

Sol:Here order is 4 as indicated. So angles symmetry can be calculated as

An = `360^0/4` = 90°.

An*1 = 90°

An*2 = 180°

An*3 = 270°

An*4 = 360°

Practice Problem figures:

Now solve the below on your own.Answers can be checked at the end in the solutions given.

Consider the following figures and calculate the angles of symmetry.

1)

The image shows order 3 rotational symmetry

2)

Star

3)

Problem figure for Rotational Symmetry

Sol:1) order = 3, A1 = 120°, A2 =240° ,A3 = 360°

2) Order = 5, A1 = 72°, A2 =144°, A3 = 216° A4 = 288° A5 = 360°

3) Order = 6, A1 = 60°, A2 =120°, A3 = 180° A4 = 240° A5 = 300° A6 = 360°.

Consider a triangular lamina ABC placed before a plane mirror MM'.You will see its image A'B'C', as shown in the figure given alongside.According to the poperties of image formed in a plane mirror, the distance of A before the mirror is same as the distance of its image A' behind the mirror i.e the mirror perpendicularly bisects the segment AA' joining the point A and its image A'.Similarly, the line segment CC', joining C and its image C', is perpendicularly bisected by the mirror aand so is the line segment BB'.

In this case,we say ,the whole figure, including triangle ABC and its image triangle A'B'C', is symmetrical about the mirror MM'.Such symmetry is called Line Symmetry.
More about Line Symmetry

A figure is said to have line symmetry if on folding the figure aboutthis line, the two parts of the figure exactly coincide.

Line symmetry of an isosceles triangle

isco sym

Consider an isosceles triangle ABC in which AB=AC.

If the triangle is folded about AX ,the bisector of angle A, two parts of triangle exactly coincide. But if the same figure is folded about BY, the bisector of angle B,the two parts do not coincide.similarly if the triangle is folded about CZ, the bisector of angle C,then also the two parts does not coincide.

In the same way if you try to fold the triangle about some other lines so that the two parts may exactly coincide, you fail to do so.

So, we conclude that an isosceles triangle has only one line of symmetry.

By rotational symmetry we mean that when an object after a certain amount of rotation looks the same,then it is said that the object has rotational symmetry. There can be more than one rotational symmetry that an object can have;.Usually, in mathematics, any rotation leavIing the object looking exactly the same, is known as rotational symmetry. In case of two dimensional objects, there are not many objects having rotationally symmetric - circles are, or sets of circles with a common center.However, there are thousands of three dimensional objects with rotational symmetry. examples spheres, cylinders, cones, tori (doughnut shapes) etc.

Learn more on about Area of a Regular Hexagon Formula and its Examples. Between, if you have problem on these topics rational numbers, Please share your comments.