**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.**

**No line of symmetry means that there is no line segment in figure which can divide**

(i) No line of symmetry :

(i) No line of symmetry :

**figure in two equal parts.**

**e.g.,**

**In letter ‘S’ there is no line of symmetry.**

**One line of symmetry means that there is only one line segment of figure which**

(ii) One line of symmetry :

(ii) One line of symmetry :

**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.**

**Two lines of symmetry means that there is two line segment which divides figure in equal parts.**

(iii) Two lines of symmetry :

(iii) Two lines of symmetry :

**e.g.,**

**In letter ‘H‘ two lines of symmetry which divide alphabet ‘H’ in equal parts.**

**There lines of symmetry means there is lines segment which divides figure in**

(iv) There lines of symmetry :

(iv) There lines of symmetry :

**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

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.

**A line has infinite length and hence it can be considered that each line perpendicular to**

(i) Symmetry of a line :

(i) Symmetry of a line :

**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.**

**A line segment has two lines of symmetry, namely, segment itself and perpendicular bisector of segment.**

(ii) Symmetry of line segment :

(ii) Symmetry of line segment :

**An angle with equal arms has one line of symmetry which is along internal bisector of angle. As OC in following figure :**

(iii) Symmetry of an angle :

(iii) Symmetry of an angle :

**(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.**

**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 :**

(vi) Symmetry of a rhombus :

(vi) Symmetry of a rhombus :

**(vii) Symmetry of a rectangle :**A rectangle has two lines of symmetry along line segment joining midpoint

**of opposite sides.**

**An arrow head, as shown in figure has diagonal BD as only line of**

(viii) Symmetry of an arrow head :

(viii) Symmetry of an arrow head :

**symmetry.**

**A semi-circle, as shown in figure has only one line of symmetry which is**

(ix) Symmetry of a semi-circle :

(ix) Symmetry of a semi-circle :

**perpendicular bisector OC of diameter AB.**

**A circle has an infinite number of lines of symmetry all along diameters.**

(x) Symmetry of a circle :

(x) Symmetry of a circle :

**A regular pentagon has five lines of symmetry as shown in following**

(xi) Symmetry of a regular pentagon :

(xi) Symmetry of a regular pentagon :

**figure.**

**A regular hexagon has six lines of symmetry. Three along lines joining**

(xii) Symmetry of a regular hexagon :

(xii) Symmetry of a regular hexagon :

**mid-point of opposite sides and three along diagonals.**

**A regular polygon of n sides n lines of symmetry.**

Remark :

Remark :

**•**

**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.**