# Chapter – 14: Symmetry

# Exercise 14.1

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

**(a)**

**Solution: –**

A figure has a line of symmetry, if there is a line about which the figure may be folded so that the two parts of the figure will coincide.

**(b)**

**Solution: –**

A figure has a line of symmetry, if there is a line about which the figure may be folded so that the two parts of the figure will coincide.

**(c)**

**Solution: –**

A figure has a line of symmetry, if there is a line about which the figure may be folded so that the two parts of the figure will coincide.

**(d)**

**Solution: –**

**(e)**

**Solution: –**

**(f)**

**Solution: –**

**(g)**

**Solution: –**

**(h)**

**Solution: –**

**(i)**

**Solution: –**

**(j)**

**Solution: –**

**(k)**

**Solution: –**

**(l)**

**Solution: –**

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

**(a)**

**Solution: –**

So, other hole is shown in the figure below.

**(b)**

**Solution: –**

So, other hole is shown in the figure below.

**(c)**

**Solution: –**

So, other hole is shown in the figure below.

**(d)**

**Solution: –**

So, other hole is shown in the figure below.

**(e)**

**Solution: –**

So, other hole is shown in the figure below.

**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)**

**Solution: –**

The concept of line of symmetry is closely related to mirror reflection. A shape has line symmetry when one half of it is the mirror image of the other half. A mirror line, thus helps to visualise a line of symmetry.

While dealing with mirror reflection, care is needed to note down the left-right changes in the orientation.

Name of the figure is square.

**(b)**

**Solution: –**

The concept of line of symmetry is closely related to mirror reflection. A shape has line symmetry when one half of it is the mirror image of the other half. A mirror line, thus helps to visualise a line of symmetry.

While dealing with mirror reflection, care is needed to note down the left-right changes in the orientation.

Name of the figure is triangle.

**(c)**

**Solution: –**

The concept of line of symmetry is closely related to mirror reflection. A shape has line symmetry when one half of it is the mirror image of the other half. A mirror line, thus helps to visualise a line of symmetry.

While dealing with mirror reflection, care is needed to note down the left-right changes in the orientation.

Name of the figure is rhombus.

**(d)**

**Solution: –**

Name of the figure is circle.

**(e)**

**Solution: –**

Name of the figure is pentagon.

**(f)**

**Solution: –**

Name of the figure is octagon.

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

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

**(a)**

**Solution: –**

Figure given has 3 lines of symmetry.

So, it has multiple lines of symmetry.

**(b)**

**Solution: –**

Figure given has 2 lines of symmetry.

So, it has multiple lines of symmetry.

**(c)**

**Solution: –**

Figure given has 3 lines of symmetry.

So, it has multiple lines of symmetry.

**(d)**

**Solution: –**

Figure given has 2 lines of symmetry.

So, it has multiple lines of symmetry.

**(e)**

**Solution: –**

Figure given has 4 lines of symmetry.

So, it has multiple lines of symmetry.

**(f)**

**Solution: –**

Figure given has only 1 line of symmetry.

**(g)**

**Solution: –**

Figure given has 4 lines of symmetry.

So, it has multiple lines of symmetry.

**(h)**

**Solution: –**

Figure given has 6 lines of symmetry.

So, it has multiple lines of symmetry.

**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: –**

By observing the above figure,

Yes, the figure will be symmetrical about both diagonals.

By observing the above figure,

Yes, the figure can be made symmetrical by more than one way.

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

**(a)**

**Solution: –**

**(b)**

**Solution: –**

**(c)**

**Solution: –**

**7. State the number of lines of symmetry for the following figures:**

**(a) An equilateral triangle**

**Solution: –**

An equilateral triangle has 3 lines of symmetry is shown in the figure below,

**(b) An isosceles triangle**

**Solution: –**

An isosceles triangle has 1 lines of symmetry is shown in the figure below,

**(c) A scalene triangle**

**Solution: –**

A scalene triangle has no line of symmetry is shown in the figure below,

**(d) A square**

**Solution: –**

A square has 4 lines of symmetry is shown in the figure below,

**(e) A rectangle**

**Solution: –**

A rectangle has 2 lines of symmetry is shown in the figure below,

**(f) A rhombus**

**Solution: –**

A rhombus has 2 lines of symmetry is shown in the figure below,

**(g) A parallelogram**

**Solution: –**

A parallelogram has no line of symmetry is shown in the figure below,

**(h) A quadrilateral**

**Solution: –**

A quadrilateral has no line of symmetry is shown in the figure below,

**(i) A regular hexagon**

**Solution: –**

A regular hexagon has 6 lines of symmetry is shown in the figure below,

**(j) A circle**

**Solution: –**

A circle has infinite lines of symmetry,

**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)** a vertical mirror

The English alphabet have reflection symmetry about a vertical mirror are, A, H, I, M, O, T, U, V, W, X, Y

**(b)** a horizontal mirror

The English alphabet have reflection symmetry about a horizontal mirror are, B, C, D, E, H, I, K, O, X

**(c)** both horizontal and vertical mirrors

The English alphabet have reflection symmetry about both horizontal and vertical mirrors are, H, I, O, X

**9. Give three examples of shapes with no line of symmetry.**

**Solution: –**

A shape has a no line of symmetry, if there is no line about which the figure may be folded and also parts of the figure will not coincide.

A scalene triangle, a quadrilateral and a parallelogram

**10. What other name can you give to the line of symmetry of**

**(a) an isosceles triangle?**

**Solution: –**

The other name to the line of symmetry of an isosceles triangle is median or altitude.

**(b) a circle?**

The other name to the line of symmetry of a circle is diameter.

## Exercise 14.2

**1. Which of the following figures have rational symmetry of order more than 1:**

**Solution: –**

**(a)**

So, the above figure has its rotational symmetry as 4.

**(b)**

So, the above figure has its rotational symmetry as 3.

**(c) **So, the given figure has only one rotational symmetry.

**(d)**

So, the above figure has its rotational symmetry as 2.

**(e)**

So, the above figure has its rotational symmetry as 3.

**(f)**

So, the above figure has its rotational symmetry as 4.

By observing all the figures (a), (b), (c), (d), (e) and (f) have rotational symmetry of order more than 1.

**2. Give the order of rotational symmetry for each figure:**

**(a)**

**Solution: –**

The above figure has its rotational symmetry as 2.

**(b)**

**Solution: –**

The above figure has its rotational symmetry as 2.

**(c)**

**(d)**

**Solution: –**

The above figure has its rotational symmetry as 4.

**(e)**

**Solution: –**

The above figure has its rotational symmetry as 4.

**(f)**

**Solution: –**

The above figure has its rotational symmetry as 5.

**(g)**

**Solution: –**

The above figure has its rotational symmetry as 6.

**(h)**

**Solution: –**

The above figure has its rotational symmetry as 3.

## Exercise 14.3

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

**Solution: –**

Equilateral triangle and Circle.

**2. Draw, wherever possible, a rough sketch of**

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

**Solution: –**

A triangle with both line and rotational symmetries of order more than 1 is an equilateral triangle.

**Line symmetry**

**Rotational symmetry**

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

**Solution: –**

A triangle with only line symmetry and no rotational symmetry of order more than 1 is isosceles triangle.

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

**Solution: –**

A quadrilateral with a rotational symmetry of order more than 1 but not a line symmetry is not possible to draw. Because a quadrilateral with a line symmetry may have rotational symmetry of order one but not more than one.

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

**Solution: –**

A quadrilateral with line symmetry but not a rotational symmetry of order more than 1 is rhombus.

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

**Solution: –**

Yes. If a figure has two or more lines of symmetry, then it will have rotational symmetry of order more than 1.

**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^{o} |

Rectangle | Intersecting point of diagonals | 2 | 180^{o} |

Rhombus | Intersecting point of diagonals | 2 | 180^{o} |

Equilateral Triangle | Intersecting point of medians | 3 | 120^{o} |

Regular Hexagon | Intersecting point of diagonals | 6 | 60^{o} |

Circle | Centre | Infinite | Every angle |

Semi-circle | Mid-point of diameter | 1 | 360^{o} |

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

**Solution: –**

The quadrilateral which have both line and rotational symmetry of order more than 1 is square.

Line symmetry:

Rotational symmetry:

**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: –**

The other angles are, 120°, 180°, 240°, 300°, 360°

So, the figure is said to have rotational symmetry about same angle as the first one. Hence, the figure will look exactly the same when rotated by 60° from the last position.

**7. Can we have a rotational symmetry of order more than 1 whose angle of rotation is**

**(i) 45°?**

**Solution: –**

Yes. We can have a rotational symmetry of order more than 1 whose angle of rotation is 45^{o}.

**(ii) 17°?**

**Solution: –**

No. We cannot have a rotational symmetry of order more than 1 whose angle of rotation is 17^{o}.

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