ICSE Class 8 Maths Chapter 31 Symmetry Reflection and Rotation

Chapter 31

Symmetry, Reflection And Rotation

31.1 Symmetry (Linear Symmetry)

Consider a plane mirror MM' and a triangle ABC placed before the mirror. As shown in the adjoining diagram, the image of triangle ABC in the mirror is triangle A'B'C'. Clearly, the images of vertices A, B and C are A', B' and C' respectively. The images of sides AB, BC and AC are A'B', B'C' and A'C' respectively. Also, the image triangle A'B'C' is congruent to the object triangle ABC.

Now, if the whole figure (including the object triangle ABC, the image triangle A'B'C' and the mirror MM') is folded about the mirror line MM'; the two parts of the figure exactly coincide i.e. A coincides with A', B with B', and C with C', similarly side AB with side A'B' and so on. Clearly, the complete figure is identical on both the sides of the line MM'. So, it (the whole figure) is said to be symmetrical about the mirror line MM'.

The line M M', about which the figure is symmetrical, is called the line of symmetry.

Fold a rectangular piece of paper as shown in figure (a). Now cut a piece of any pattern from the folded side of the paper as shown in figure (b).

On unfolding the cutting, a design as shown in figure (c) is obtained. It is clear from the figure that the design obtained is identical on both the sides of the crease of the paper, as shown by the dotted line CD. If the figure (c) is folded again about the line CD in it, the two parts of the figure will exactly coincide. So, we say that the figure (c) is symmetrical about the line CD in it. Here, line CD is the line of symmetry.

A plane figure is said to have symmetry (or linear symmetry) if on folding the figure about a line on it, the two parts of the figure exactly coincide.

The line, about which the figure is symmetrical is called a line of symmetry or an axis of symmetry or simply a mirror line.

31.2 Examples

In each of the following figures, the dotted lines are the lines of symmetry.

A line segment is symmetrical about its perpendicular bisector.

An angle with equal arms is symmetrical about its bisector.

An isosceles triangle is symmetrical about the bisector of the angle of vertex.

The bisector of the angle of vertex of an isosceles triangle is also the perpendicular bisector of its base.

An equilateral triangle is symmetrical about the bisector of each angle of vertex, so it has three lines of symmetry.

In this case also, each angle bisector is the perpendicular bisector of the opposite side.

Each of these figures has one line of symmetry.

Each figure has two lines of symmetry.

A regular polygon: The number of lines of symmetry equals the number of sides in the polygon.

Equilateral triangle: Three lines of symmetry

Square: Four lines of symmetry

Regular pentagon: Five lines of symmetry

Regular hexagon: Six lines of symmetry

Letters of English alphabet:

Only one line of symmetry

Two lines of symmetry

A circle: Any line which passes through the centre is a line of symmetry. A circle has an infinite number of lines of symmetry.

Teacher's Note

Symmetry is visible in nature everywhere - from butterfly wings to flower petals to snowflakes. Recognizing symmetry helps us understand natural patterns and appreciate the mathematical beauty in the world around us.

Test Yourself

If on folding the given figure about the line PQ; the two parts of the figure exactly coincide; then PQ is called ... and the whole figure is said to be ...

A quadrilateral with:

(a) only one line of symmetry is ...

(b) exactly two lines of symmetry is ...

(c) exactly three lines of symmetry is ...

(d) exactly four lines of symmetry is ...

A polygon with 8 sides may have at the most ... lines of symmetry.

A polygon may have no line of symmetry. Is this statement true? ...

If true, state names of two polygons which do not have any line of symmetry.

1. ... 2. ...

Draw three capital letters of English alphabet which have exactly two lines of symmetry.

1. ... 2. ... 3. ...

Exercise 31 (A)

State, whether true or false:

(i) The letter B has one line of symmetry.

(ii) The letter F has no line of symmetry.

(iii) The letter O has only two lines of symmetry.

(iv) The figure ... has no line of symmetry.

(v) The letter N has one line of symmetry.

(vi) The figure ... has one line of symmetry.

(vii) The letter D has only one line of symmetry

(viii) A scalene triangle has three lines of symmetry.

If possible, draw the largest number of lines of symmetry in each case:

(i) A diamond shape

(ii) A flower shape

(iii) A flower shape

(iv) A circle with center marked

(v) Two circles

(vi) A parallelogram

Examine each of the following figures, carefully, and then draw lines of symmetry if possible:

(i) A right triangle

(ii) A right triangle

(iii) A triangle with equal marks

(iv) A quadrilateral with marks

(v) Two circles

(vi) A rounded rectangle

Draw all lines of symmetry for each of the following letters:

(i) C

(ii) E

(iii) F

(iv) K

(v) X

(vi) V

Construct a triangle ABC in which AB = AC = 5 cm and BC = 6 cm. Draw all its lines of symmetry.

Construct a triangle PQR in which: QR = 4.6 cm. angle Q = angle R = 50 degrees. Draw all its lines of symmetry.

Construct a triangle XYZ in which: XY = YZ = ZX = 4.5 cm. Draw all its lines of symmetry.

Construct a triangle ABC in which: AB = BC = 4 cm and angle ABC = 60 degrees. Draw all its lines of symmetry.

Construct a triangle PQR in which: PQ = QR = 4.2 cm and angle PQR = 90 degrees. Draw all its lines of symmetry.

Mark two points A and B 6-4 cm. apart. Construct the line of symmetry so that the points A and B are symmetric with respect to this line.

Mark two points P and Q 5.3 cm. apart. Construct the perpendicular bisector of line segment PQ. Are the points P and Q symmetric with respect to the perpendicular bisector drawn?

Teacher's Note

Creating symmetrical figures through construction teaches precision and helps students visualize how geometric principles create balanced, aesthetically pleasing designs used in architecture and art.

31.3 Reflection

Consider a plane mirror M M' and an object P before it. The image of object P will be formed at point P' such that:

(i) the size of image P' is same as the size of object P.

(ii) the distance of object P before the mirror is same as the distance of image P' behind the mirror.

(iii) the mirror M M' is perpendicular bisector of the line segment P P', joining the object P and its image P'.

Reflection of a point in x-axis: Consider a point P(x, y). In order to get its reflection in x-axis; draw PA perpendicular to x-axis and produce upto P' such that PA = P'A.

We observe the co-ordinates of image point P' = (x, -y)

Reflection of P(x, y) in x-axis = P'(x, -y).

When a point is reflected in x-axis, the sign of its y-co-ordinate (ordinate) changes.

Reflection of a point in y-axis: Consider a point P(x, y). In order to get its reflection in y-axis, draw PB perpendicular to y-axis and produce upto P' such that PB = P'B.

We observe the co-ordinates of image point P' = (-x, y)

Reflection of P(x, y) in y-axis = P'(-x, y).

When a point is reflected in y-axis, the sign of its x-co-ordinate (abscissa) changes.

Reflection of a point in origin: Consider a point P(x, y). In order to get its reflection in origin O, join PO and produce upto point P' such that PO = P'O.

We observe the co-ordinates of image point P' = (-x, -y)

Reflection of P(x, y) in origin = P'(-x, -y).

When a point is reflected in origin, signs of its x-co-ordinate (abscissa) and y-co-ordinate (ordinate) both change.

Test Yourself

Teacher's Note

Reflections are used in everyday life from mirrors in our homes to designing symmetrical logos in business. Understanding how coordinates change under reflection helps in computer graphics and design applications.

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