Chapter-14 SYMMETRY

Solution :

(a) In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. The axes of symmetry is shown below.

(b) In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. The axes of symmetry is shown below.

(c) In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. The axes of symmetry is shown below.

(d) In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. The axes of symmetry is shown below.

(e) line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. The different axes of symmetry are shown below:

(f) In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. The axes of symmetry is shown below.

(g) In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. The axes of symmetry is shown below.

(h) In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. The axes of symmetry is shown below.

(i) In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. The axes of symmetry is shown below.

(j) In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. The axes of symmetry is shown below

(k) In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. The different axes of symmetry are shown below.

(l) In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. The axes of symmetry is shown below.

Solution:

(a) In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. 

In the figure, the other hole is shown according to line of symmetry.

(b) In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another.

In the figure, the other hole is shown according to line of symmetry.

(c) In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another.

 In the figure, the other hole is shown according to line of symmetry.

(d) In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. 

(e) In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. 

In the figure, the other hole is shown according to line of symmetry.

Solution:

a)In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. Visualising a mirror line will be helpful.

Mirror reflection should be formed carefully by considering different orientation and possibilities.

Figure obtained is called Square.

(b) In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. Visualising a mirror line will be helpful.

Mirror reflection should be formed carefully by considering different orientation and possibilities.

Figure obtained is called Triangle.

(c)In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. Visualising a mirror line will be helpful.

Mirror reflection should be formed carefully by considering different orientation and possibilities.

Figure obtained is called Rhombus.

(d)In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. Visualising a mirror line will be helpful.

Mirror reflection should be formed carefully by considering different orientation and possibilities.

Figure obtained is called Circle.

(e) In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. Visualising a mirror line will be helpful.

Mirror reflection should be formed carefully by considering different orientation and possibilities.

Figure obtained is called Pentagon.

(f)In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. Visualising a mirror line will be helpful.

Mirror reflection should be formed carefully by considering different orientation and possibilities.

Figure obtained is called Octagon.

Solution:

a)There are 3 lines of symmetries.

Hence, it contains multiple line of symmetry.

(b) There are 2 lines of symmetries.

Hence it contains multiple line of symmetry.

(c)

There are 3 lines of symmetries.

Hence, it contains multiple lines of symmetries.

(d)There are 2 lines of symmetries.

Hence, it contains multiple lines of symmetries.

(e)There are 4 lines of symmetries.

Hence, it contains multiple lines of symmetries.

Solution:

Yes, the above given shape can be made symmetrical about both the diagonal axes by shading the proper boxes.

Yes, the above given shape can be made symmetrical about more than one axes by shading the proper boxes.

Solution:

In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. Visualising a mirror line will be helpful.

Solution:

In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. 

The equilateral triangle contains 3 lines of symmetry as shown below.

(b) In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. 

The isosceles triangle contains 1 line of symmetry as shown below.

(c) In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. 

The scalene triangle contains 0 lines of symmetry as shown below.

(d) In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. 

The square contains 4 lines of symmetry as shown below.

(e) In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. 

The rectangle contains 2 lines of symmetry as shown below.

(f)In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. 

The rhombus contains 2 lines of symmetry as shown below.

(g) In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. 

The parallelogram contains 0 lines of symmetry as shown below.

(h) In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. 

The quadrilateral contains 0 lines of symmetry as shown below.

(i) In line symmetry, if you were to fold the object or figure along the line, the two halves would match up perfectly or they will be mirror to one another. 

The regular hexagon contains 6 lines of symmetry as shown below.

Solution:

(a) a vertical mirror

A, H, I, M, O, T, U, V, W, X, Y - these alphabets of English have mirror reflection about vertical axes.

(b) a horizontal mirror

B, C, D, E, H, I, K, O, X - these alphabets of English have mirror reflection about horizontal axes.

Solution:

A scalene triangle, a quadrilateral and a parallelogram, these shapes do not have any line of symmetry. So, they cannot be fold in two symmetric halves.

Solution:

(a) Altitude or Median is the other name of the line of symmetry in isosceles triangle.

(b) Diameter is another name of the line of symmetry in a circle.

Solution-

(a) 

Number of symmetries of rotation is 4 as shown above. All are 90o apart.

(b)

Number of symmetries of rotation are 3 as shown above. All are 120o apart.

(c)

Number of symmetries of rotation is only 1 as shown above.

(d)

Number of symmetries of rotation are 2 as shown above. At 180o apart.

(e)

Number of symmetries of rotation is 3 as shown above. At 120o apart.

(f) 

Number of symmetries of rotation are 4 as shown above. At 90o apart.

Solution-

(a)

The order of rotational symmetry is 2 here.

(b)

The order of rotational symmetry is 2 here.

(c)

The order of rotational symmetry is 3 here. At 120o apart.

(d)

The order of rotational symmetry is 4 here. At 90o apart.

(e)

The order of rotational symmetry is 4 here. At 90o apart.

(f)

The order of rotational symmetry is 5 here. At 72o apart.

(g)

The order of rotational symmetry is 6 here. At 60o apart.

(h)

The order of rotational symmetry is 3 here. At 120o apart.

Solution:

Circle and Square.

Solution:

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

Line -

Rotational –

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

(iii)

It is impossible to draw a quadrilateral which have rotational symmetry of order more than one but do not have any line symmetry because if a quadrilateral has rotational symmetry of order more than 1, it means that it can be rotated by an angle and still look the same, and it can also be rotated by some multiple of that angle and still look the same. 

If a quadrilateral has no line symmetry, it means that there is no line that can divide the quadrilateral into two congruent halves.

However, if a quadrilateral has rotational symmetry of order more than 1, it means that it has at least one rotational axis of symmetry. This rotational axis of symmetry is also a line of symmetry, because if you reflect the figure across this line and then rotate it by the same angle, it will look the same.

(iv)

A Rhombus is a quadrilateral with no rotational symmetry but line symmetry with order more than 1.

Solution:

Yes, if a figure has two or more lines of symmetry, it must have rotational symmetry of order more than 1. This is because each line of symmetry corresponds to a rotation of the figure around a certain point.

Solution-

Solution:

The figure will look exactly the same at angles of, 120°, 180°, 240°, 300°, 360°.

As these above angles are multiple of 60°, so rotating the figure by above angles wouldn’t change or it will look the same.

Solution:

(i) Yes. The rotational symmetry of order more than 1 is possible with angle of rotation as 45o. Angle 360o is a multiple of 45o. 

(ii) No. The rotational symmetry of order more than 1 is not possible with angle of rotation as 17o. Angle 360o is not a multiple of 17o.