# 6 Math Chapter 13 Symmetry

Chapter Notes and Summary
Line of symmetry : A figure is said to be line of symmetry, if by folding figure along a line, left and right parts of it coincide exactly. This line is called line (or axis) of symmetry of figure.
• In given figure, l is line of symmetry because it divides left and right parts of figure equally.
A figure contains many types of line of symmetry, such as, no line of symmetry, one line of symmetry, two lines of symmetry and multiple (or more than two) lines of symmetry.
(i) No line of symmetry :
No line of symmetry means that there is no line segment in figure which can divide figure in two equal parts.
e.g., In letter ‘S’ there is no line of symmetry.
(ii) One line of symmetry :
One line of symmetry means that there is only one line segment of figure which divides figure in two equal parts of figure.
In letter ’T’ there is one line of symmetry which divides alphabet ‘T’ in two equal parts.
(iii) Two lines of symmetry :
Two lines of symmetry means that there is two line segment which divides figure in equal parts.
e.g., In letter ‘H‘ two lines of symmetry which divide alphabet ‘H’ in equal parts.
(iv) There lines of symmetry :
There lines of symmetry means there is lines segment which divides figure in equal parts.
e.g., In an equilateral triangle, there are three medians (AD, BE and CF) in triangle (ABC) which divides triangle in equal parts.
Hence, line symmetry of equilateral triangle is three.
Figures with more than two lines of symmetry : In this section, we will discuss some figures having more than two lines of symmetry. Let us take a square piece of paper. Fold it into half vertically and then fold it again into half horizontally. Open out folds. You will get two lines of symmetry, one horizontal and one vertical. Now, fold paper into along a diagonal. Open it and fold it into half along water diagonal. Open out fold.
Now you will get two more lines of symmetry one along each diagonal. Thus, a square has four lines of symmetry as shown below :
First fold Second fold Third fold Fourth fold
Symmetry of some figure : Let us now discuss symmetry of some standard geometrical figures.
(i) Symmetry of a line :
A line has infinite length and hence it can be considered that each line perpendicular to given line divide line into two equal halves (parts). So, a line has infinite number of symmetrical lines which are perpendicular to it. Also a line is symmetrical to itself.
(ii) Symmetry of line segment :
A line segment has two lines of symmetry, namely, segment itself and perpendicular bisector of segment.
(iii) Symmetry of an angle :
An angle with equal arms has one line of symmetry which is along internal bisector of angle. As OC in following figure :
(iv) Symmetry of an isosceles triangle : As isosceles triangle has one line of symmetry which is along median through vertex. Median AD is a line of symmetry of isosceles triangle ABC, as AB = AC in following figure :
(v) Symmetry of parallelogram : A parallelogram has no line of symmetry. In following figure parallelogram ABCD has no line of symmetry.
(vi) Symmetry of a rhombus :
A rhombus has two lines of symmetry along its diagonals. In rhombus ABCD, diagonals AC and BD are two lines symmetry. As shown following figure :
(vii) Symmetry of a rectangle : A rectangle has two lines of symmetry along line segment joining midpoint of opposite sides.
(viii) Symmetry of an arrow head :
An arrow head, as shown in figure has diagonal BD as only line of symmetry.
(ix) Symmetry of a semi-circle :
A semi-circle, as shown in figure has only one line of symmetry which is perpendicular bisector OC of diameter AB.
(x) Symmetry of a circle :
A circle has an infinite number of lines of symmetry all along diameters.
(xi) Symmetry of a regular pentagon :
A regular pentagon has five lines of symmetry as shown in following figure.
(xii) Symmetry of a regular hexagon :
A regular hexagon has six lines of symmetry. Three along lines joining mid-point of opposite sides and three along diagonals.
Remark :
A regular polygon of n sides n lines of symmetry.
Reflection and Symmetry : Line symmetry is closely related to mirror reflection. distance of image of a point from line of symmetry is same as that of point from that line of symmetry.
A figure is said to be symmetrical about a line l, if it is identical an either side of l, where l is called line of axis of symmetry.
• English Alphabets having symmetry : Each of following capital letters of English alphabet is symmetrical about dotted line or lines as shown given below :
Application in Everyday Life : Symmetry has plenty of application in everyday life as in art, architecture, textile technology, design creations, geometrical reasoning Kolams, Rangoli etc.