Class 6 – Symmetry

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Symmetry
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Introduction: Observe the following images (figures)

If the above images are folded over the line, what happens?

Here the images (i), (ii), (iv), (vii) are over lapping. Part of images on left side of line are equal to the part of image on right side (identical).

∴ If you fold a picture in half such that the left and right halves match exactly then the picture is said to have symmetry with respect of line. The example below shows a few more shapes which are identical on either side of the line.

Example:

Symmetry is an important geometrical concept, commonly exhibited in nature and is used almost in every field of activity. Artist’s, designers of clothing (or) jewellery, vehicle manufacturers and architects make use of the idea of symmetry.

http://images.clipartpanda.com/bloc-clipart-note-md.pngSymmetry is used to balance (proportion) the structure of human beings, buildings, vehicle’s and so on….

 

Line symmetry: If there is a line about which the figure may be folded so that the two parts of the figure will coincide the line is a line of symmetry or an axis of symmetry.

Example:

Example: Which of the following figures have line of symmetry?

Solution: Here the figure (i) has line symmetry The image on left sides of line is identical with right side and other two figure are not same on the other part.

Reflection: The process of getting the image of an object with respect to a line is called reflection.

Example: Sita painted a picture in her note book she closed the note book before the painting was dry later she opened it and, saw her painting on the opposite page.

Observe the following and identify the pair of figures which are reflected.

Observation: (i) and (iii) are reflected with respect of line.

Reflection of shapes: You can reflect in a mirror to make another shape.

Reflection line: Experiments in physics show that if  is the image of a upon reflection in the mirror m, then the line m is the perpendicular bisector of the line .

The line m is called the mirror line or the reflection line. This line is also called the line of symmetry.

If  is the image of A after reflection in the line m, then the point A is called the pre image of the point .

http://images.clipartpanda.com/bloc-clipart-note-md.pngA reflection in a line produces a mirror image. The image is the same size but it is a reflection of the original shape.

The above shapes are symmetrical about the dotted line. Thus dotted line is the axis of

Symmetry. One half of the shape is the mirror image of the other half.
If we imagine a mirror placed along the axis of the symmetry, the image in the mirror would look exactly like the hidden half with the half shell showing, we can still see the original shape.
Also, if folded the figure in half along the line of symmetry or mirror line will fall exactly on one another.

A figure may have one line symmetry, two lines of symmetry, three or more lines of symmetry or no line of symmetry.

Here, (i) has one line of symmetry

(ii) have two lines of symmetry

(iii) have four lines of symmetry

(iv) has no line of symmetry.

Symmetry of alphabets: The following are some of the capital letters of English alphabet that are symmetrical about the dotted line, which is their axis of symmetry.

If we observe these figures we can say that

i)    The letters C and D have horizontal line of symmetry.

ii)   The letters A, U and M have vertical line of symmetry.

iii) The letter H has both (horizontal and vertical) lines of symmetry.

iv)  The letter O has infinite lines of symmetry.

 

Symmetry in plane figures: Shown below are some geometrical figures with their lines of symmetry.

Figure

Name of the figure

Number of lines of symmetry

Equilateral Triangle

Three

Square

Four

Rectangle

Two

Isosceles

Triangle

One

 

Rhombus

Two

Circle

Infinite (all the diameters)

Hexagon

Six

http://www.sandrila.co.uk/articles/visio-articles/visio-flowchart-shapes/images/data.png

      Parallelogram

Zero

 

Example: Find number of lines of symmetry in an Isosceles triangle.

Solution: A nisosceles triangle has one line of symmetry, which is the perpendicular bisector of unequal side.

Example: How many lines of symmetry does a scalene triangle have?

Solution: A scalene triangle has no line of symmetry.

 

Example: How many lines of symmetry does a square have?

Solution: A square has four lines of symmetry, which are the diagonals and line joining mid points of opposite sides.

Example: How many lines of symmetry does a rectangle have?

Solution: A rectangle has two lines of symmetry, which are the line joining the midpoints of opposite side.

Example: Find number of lines of symmetry for a regular pentagon.

Solution: A pentagon has five lines of symmetry, which are the perpendicular bisectors of the sides.

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Reflection Symmetry: The simplest symmetry is Reflection Symmetry (sometimes called Line Symmetry or MirrorSymmetry). It is easy to see, because one half is the reflection of the other half.

http://www.mathsisfun.com/geometry/images/flame-mirror.jpg

Here my dog “Flame”has her face made perfectly symmetrical with a bit of photo magic. The white line down the center is the Line of Symmetry.

http://www.mathsisfun.com/geometry/images/reflection-parinacota.jpg

The reflection in this lake also has symmetry, but in this case:

The Line of Symmetry runs left-to-right it is not perfect symmetry, as the image is changed a little by the lake surface.

The Line of Symmetry can be in any direction (notjust up-down or left-right).