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Name any two figures that have both line symmetry and rotational symmetry.
Equilateral triangle and regular hexagon have both line of symmetry and rotational symmetry.
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.
(i) Equilateral triangle has 3 lines of symmetry and rotational symmetry of
order 3.
(ii) Isosceles triangle has only 1 line of symmetry and no rotational symmetry of order more than 1.
(iii) A parallelogram is a quadrilateral which has no line of symmetry but a rotational symmetry of order 2.
(iv)A kite is a quadrilateral which has only 1 line of symmetry and no rotational symmetry of order more than 1.
If a figure has two or more lines of symmetry, should it have rotational symmetry of order more than 1?
Yes. If a figure has two or more lines of symmetry, then it will definitely have its rotational symmetry of order more than 1.
Fill in the blanks:
The given table can be completed as follows.
| Shape | Centre of Rotation | Order of Rotation | Angle of Rotation |
| Square | Intersection point of diagonals | 4 | 90º |
| Rectangle | Intersection point of diagonals | 2 | 180º |
| Rhombus | Intersection point of diagonals | 2 | 180º |
| Equilateral Triangle | Intersection point of medians | 3 | 120º |
| Regular Hexagon | Intersection point of diagonals | 6 | 60º |
| Circle | Centre | Infinite | Any angle |
| Semi-circle | Centre | 1 | 360º |
Name the quadrilaterals which have both line and rotational symmetry of order more than 1.
Square, rectangle, and rhombus are the quadrilaterals which have both line and rotational symmetry of order more than 1. A square has 4 lines of symmetry and rotational symmetry of order 4. A rectangle has 2 lines of symmetry and rotational symmetry of order 2. A rhombus has 2 lines of symmetry and rotational symmetry of order 2.
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?
It can be observed that if a figure looks symmetrical on rotating by 60º, then it will also look symmetrical on rotating by 120º, 180º, 240º, 300º, and 360º i.e., further multiples of 60º.
Can we have a rotational symmetry of order more than 1 whose angle of rotation is
(i) 45°?
(ii) 17°?
It can be observed that if the angle of rotation of a figure is a factor of 360º, then it will have a rotational symmetry of order more than 1.
It can be checked that 45º is a factor of 360º but 17º is not. Therefore, the figure having its angle of rotation as 45º will have its rotational symmetry of order more than 1. However, the figure having its angle of rotation as 17º will not be having its rotational symmetry of order more than 1.
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