Symmetry Formula

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Symmetrical Meaning

Symmetry is the most common and interesting topic in mathematics. When a figure can be divided into two halves that are exact reflections of each other, as shown in Figure 1, such figures are said to be symmetrical. These figures have line symmetry. If we were to fold each figure in half at the red lines, the two halves would lie exactly on top of each other. And the red line is said to be a line of symmetry or axis of symmetry. One shape becomes exactly like another shape when we turn it, flip it, or slide it. For two objects to be symmetrical, it is necessary that they must be the same size and shape.

And the shapes which are not identical on both sides of the centerline are called asymmetrical shapes.

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Let us study symmetrical meaning and types of symmetry in detail.


Line of Symmetry

The figure is divided along the centerline which is an imaginary line this line is called the axis of symmetry or the line of symmetry. The line of symmetry can be horizontal, vertical, or diagonal. There can be one, two, or more lines of symmetry. From the below figure the rectangle is divided horizontally and vertically. The dashed line is called the line of symmetry. There can be two lines of symmetry in a rectangle.

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  • Example of a figure having one line of symmetry

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  • Example of a figure having two lines of symmetry.

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  • Some figures have an infinite number of lines of symmetry. The best example is a circle with an infinite number of lines of symmetry.

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Types of Symmetry

Symmetry in Maths is divided into several types of symmetry. They are:

 Types of symmetry are as follows:

  • Reflection Symmetry

  • Rotational Symmetry

  • Point Symmetry

  • Translational Symmetry


Reflection Symmetry

When a figure is divided into two identical halves such that one is a mirror image of the other then it is called reflection symmetry. It is the simplest type of symmetry often referred to as line symmetry or mirror symmetry.

Example:

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Rotational Symmetry  

A figure that looks exactly the same after rotation as it was before rotation is called rotational symmetry. When an object is made to rotate, its size and shape do not change. Rotation can be clockwise or anticlockwise.

The fixed point about which the object is rotated is called the center of rotation.

Example :

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Point Symmetry

When every part of a figure has a matching part in the opposite direction from the central point it is said to have point symmetry. Point symmetry is sometimes called ‘Origin Symmetry’. Because the center point is the origin where the figure is symmetrical.

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Order of Rotational Symmetry

The number of distinct orientations in which the shape looks the same as the original is called its order of rotational symmetry.

A full turn means a rotation of about 3600. In a full turn, there are mainly 4 rotational positions

  • Rotation through 900, which is called a quarter turn.

  • Rotation through 1800, which is called half turn.

  • Rotation through 2700, which is called a three - fourth turn.

  • Rotation through 3600, which is called a full turn.


Axis of Symmetry

Axis of symmetry is a line that divides an object into two equal halves, which creates a mirror-like reflection of either side of the object..

To find the axis of symmetry there are two different formulas that you can use. One is when the parabola's equation is in vertex form  and one is  when the parabola's equation is in standard form  .

  • If your equation is in the standard form we have

y = ax2 + bx + c

Where,

a and b are coefficients of x2 and x respectively.

then the formula for the axis of symmetry is given as below

x = −b / 2a

  • If your equation is in vertex form 

y = (x−h)2 + k

y = (x−h)2 + k ,

then the formula for axis of symmetry is:

x = h


Solved Examples

Example 1:

Find the axis of symmetry of the graph of y = 4x2 + 12x - 7. 

Solution:

y = 4x2 + 12x -7

We have standard form as

y = ax2 + bx + c,

Comparing the coefficient of a and b

a = 4, 

b = 12, 

c = -7

Formula for axis of symmetry x = -b/2a

Substituting the values of a and b,

x = -12/2(4)

= -12/8

= -3/2

Hence, the axis of symmetry is x = -3/2

FAQ (Frequently Asked Questions)

1. How Symmetry is Used in Daily Life?

We observe symmetry all around us. Symmetry means a shape or an object when divided from the center and has two identical halves.  We can say that both halves are mirror images of each other. Symmetry makes the object more beautiful. We observe symmetry in nature like flowers, trees, butterflies, some insects, animals, etc. We humans also observe symmetry in most of the cases. More symmetrical is the person,  more attractive the person is. The human body organs also have a line of symmetry like the kidney, lungs, etc. Even various man-made art are symmetrical like buildings, monuments, statues, etc.

2. Mention the Real-Life Examples of Symmetry

  • Reflection of trees in clear water.

  • Reflection of mountains in a lake.

  • Butterflies are identical on both sides.

  • Some human faces are identical on the left and right sides.

  • People can also have an identical mustache on both sides.