Chapter 14: Symmetry

 

Exercise: 14.1 (10)

 

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

Answer figures are:

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.   

Solution

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

(h) A quadrilateral

(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.

 

 

Exercise: 14.2 (2)

 

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

 

 

Exercise: 14.3 (7)

 

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?

Solution

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° MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaGaaGOnaiaaic dacqGHWcaSaaa@393C@  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° MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaGaaGOnaiaaic dacqGHWcaSaaa@393C@  then it will also looks symmetrical on rotating by 120°,180°,240°,300°,360° MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaik dacaaIWaGaeyiSaaRaaiilaiaaigdacaaI4aGaaGimaiabgclaWkaa cYcacaaIYaGaaGinaiaaicdacqGHWcaScaGGSaGaaG4maiaaicdaca aIWaGaeyiSaaRaaiilaiaaiodacaaI2aGaaGimaiabgclaWcaa@4D36@  

For 60° MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaGaaGOnaiaaic dacqGHWcaSaaa@393C@  rotation:

It will rotate six times.

For 120° MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaik dacaaIWaGaeyiSaalaaa@39F3@  rotation:

It will rotate three times.

For 180° MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaiI dacaaIWaGaeyiSaalaaa@39F9@  rotation:

It will rotate two times.

For 360° MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaaiA dacaaIWaGaeyiSaalaaa@39F9@  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° MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaGaaGinaiaaiw dacqGHWcaSaaa@393F@ ?

(ii)          17° MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaiE dacqGHWcaSaaa@393E@ ?

Solution

(i)             It can be observed that if the angle of rotation of a figure is a factor of 360°, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaaiA dacaaIWaGaeyiSaaRaaiilaaaa@3AA9@  then it will have rotational symmetry of order more than one.

If the angle of rotation is 45°, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaGaaGinaiaaiw dacqGHWcaScaGGSaaaaa@39EF@  then symmetry of order is possible and would be 8 rotations.

(ii)          If the angle of rotational is 17°, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaiE dacqGHWcaScaGGSaaaaa@39EE@  then symmetry of order is not possible because 360° MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaaiA dacaaIWaGaeyiSaalaaa@39F9@  is not completely divided by 17° MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaiE dacqGHWcaSaaa@393E@ .