Chapter 14: Symmetry

# Question: 1

Copy the figures with punched holes and find the axes of symmetry for the following:

(a)                   (b)                   (c)                   (d)

(e)                   (f)                   (g)                   (h)

(i)                    (j)                   (k)                    (l)

## Solution

 S.No. Punched holed figures The axes of symmetry (a) (rectangle) (b) (square) (c) (d) (e) (square) (f) (g) (h) (i) (j) (k) (l)

# Question: 2

Given the line(s) of symmetry, find the other hole(s):

(a)             (b)               (c)             (d)               (e)

## Solution

 S.No. Line(s) of symmetry Other holes on figures (a) (b) (c) (d) (e)

# Question: 3

In the following figures, the mirror line (i.e., the line of symmetry) is given as a dotted line. Complete each figure performing reflection in the dotted (mirror) line. (You might perhaps place a mirror along the dotted line and look into the mirror for the image). Are you able to recall the name of the figure you complete?

(a)                (b)                  (c)                   (d)                  (e)                   (f)

## Solution

 S.No. Question figures Complete figures Names of the figure (a) Square (b) Triangle (c) Rhombus (d) Circle (e) Pentagon (f) Octagon

# Question: 4

The following figures have more than one line of symmetry. Such figures are said to have multiple lines of symmetry.

(a)                                      (b)                                 (c)

Identify multiple lines of symmetry, if any, in each of the following figures:

(a)                         (b)                      (c)                            (d)

(e)                          (f)                       (g)                            (h)

## Solution

 S.No. Problem Figures Lines of symmetry (a) (b) (c) (d) (e) (f) (g) (h)

# Question: 5

Copy the figure given here.

Take any one diagonal as a line of symmetry and shade a few more squares to make the figure symmetric about a diagonal. Is there more than one way to do that? Will the figure be symmetric about both the diagonals?

## Solution

Yes, there is more than one way.

Yes, this figure will be symmetric about both the diagonals.

# Question: 6

Copy the diagram and complete each shape to be symmetric about the mirror line(s):

a.

b.

c.

a.

b.

c.

# Question: 7

State the number of lines of symmetry for the following figures:

(a) An equilateral triangle

(b) An isosceles triangle

(c) A scalene triangle

(d) A square

(e) A rectangle

(f) A rhombus

(g) A parallelogram

(i) A regular hexagon

(j) A circle

## Solution

 S.No. Figure’s name Diagram with symmetry Number of lines of symmetry (a) Equilateral triangle 3 (b) Isosceles triangle 1 (c) Scalene triangle 0 (d) Square 4 (e) Rectangle 2 (f) Rhombus 2 (g) Parallelogram 0 (h) Quadrilateral 0 (i) Regular Hexagon 6 (j) Circle Infinite

# Question: 8

What letters of the English alphabet have reflectional symmetry (i.e., symmetry related to mirror reflection) about.

a.    a vertical mirror

b.   a horizontal mirror

c.    both horizontal and vertical mirrors

## Solution

a.    Vertical mirror – A, H, I, M, O, T, U, V, W, X and Y

b.    Horizontal mirror – B, C, D, E, H, I, K, O and X

c.    Both horizontal and vertical mirror – H, I, O and X

# Question: 9

Give three examples of shapes with no line of symmetry.

## Solution

The three examples are:

a.    A trapezium

b.   A scalene triangle, and

c.    A parallelogram

# Question: 10

What other name can you give to the line of symmetry of

a.    an isosceles triangle?

b.   a circle?

## Solution

a.    The line of symmetry of an isosceles triangle is also called median or altitude.

b.   The line of symmetry of a circle is the diameter of the circle.

# Question: 1

Which of the following figures have rotational symmetry of order more than 1:

(a)             (b)             (c)              (d)                  (e)               (f)

## Solution

Rotational symmetry of order more than 1 are (a), (b), (d), (e) and (f) because in these figures, after complete turn, more than 1 number of times, an object looks exactly the same.

# Question: 2

Give the order of rotational symmetry for each figure:

(a)                    (b)                    (c)                   (d)

(e)               (f) (g)                  (h)

## Solution

 S.No. Problem Figures Rotational figures Order of rotational symmetry (a) 2 (b) 2 (c) 3 (d) 4 (e) 4 (f) 5 (g) 6 (h) 3

# Question: 1

Name any two figures that have both line symmetry and rotational symmetry.

## Solution

A circle and an equilateral triangle have both line symmetry and rotational symmetry.

# Question: 2

Draw, wherever possible, a rough sketch of

(i)             a triangle with both line and rotational symmetries of order more than 1.

(ii)          a triangle with only line symmetry and no rotational symmetry of order more than 1.

(iii)      a quadrilateral with a rotational symmetry of order more than 1 but not a line symmetry.

(iv)       a quadrilateral with line symmetry but not a rotational symmetry of order more than 1.

## Solution

(i)             An equilateral triangle has both line and rotational symmetries of order more than 1.

Line symmetry:

Rotational symmetry:

(ii)          An isosceles triangle has only one line of symmetry and no rotational symmetry of order more than 1.

Line symmetry:

Rotational symmetry:

(iii)       A parallelogram is a quadrilateral which has no line of symmetry but a rotational symmetry of order 2.

(iv)       A trapezium which has equal non-parallel sides, is a quadrilateral with line symmetry but not a rotational symmetry of order more than 1.

Line symmetry:

Rotation  al symmetry:

# Question: 3

If a figure has two or more lines of symmetry, should it have rotational symmetry of order more than 1?

Yes

# Question: 4

Fill in the blanks:

 Shape Centre of Rotation Order of Rotation Angle of Rotation Square Rectangle Rhombus Equilateral Triangle Regular Hexagon Circle Semi-circle

## Solution

 Shape Centre of Rotation Order of Rotation Angle of Rotation Square Intersecting point of diagonals. 4 90° Rectangle Intersecting point of diagonals. 2 180° Rhombus Intersecting point of diagonals. 2 180° Equilateral Triangle Intersecting point of medians. 3 120° Regular Hexagon Intersecting point of diagonals. 6 60° Circle Centre infinite At every point Semi-circle Mid-point of diameter 1 360°

# Question: 5

Name the quadrilaterals which have both line and rotational symmetry of order more than 1.

## Solution

Square has both line and rotational symmetry of order more than 1.

Line symmetry:

Rotational symmetry:

# Question: 6

After rotating by $60°$ about a centre, a figure looks exactly the same as its original position. At what other angles will this happen for the figure?

## Solution

It can be observed that if a figure looks symmetrical after rotating about $60°$ then it will also looks symmetrical on rotating by $120°,180°,240°,300°,360°$

For $60°$ rotation:

It will rotate six times.

For $120°$ rotation:

It will rotate three times.

For $180°$ rotation:

It will rotate two times.

For $360°$ rotation:

It will rotate one time.

# Question: 7

Can we have a rotational symmetry of order more than 1 whose angle of rotation is:

(i)             $45°$?

(ii)          $17°$?

## Solution

(i)             It can be observed that if the angle of rotation of a figure is a factor of $360°,$ then it will have rotational symmetry of order more than one.

If the angle of rotation is $45°,$ then symmetry of order is possible and would be 8 rotations.

(ii)          If the angle of rotational is $17°,$ then symmetry of order is not possible because $360°$ is not completely divided by $17°$.