Exercise 14.3 Symmetry -NCERT Solutions Class 7
Exercise 14.3
Chapter 14 Ex.14.3 Question 1
Name any two figures that have both line symmetry and rotational symmetry.
Solution
What is unknown?
The figures that have both line symmetry and rotational symmetry.
Reasoning:
To solve this question, remember the concept of line symmetry and rotational symmetry. The ‘Line of Symmetry’ is the imaginary line where you could fold the image and have both halves match exactly. Rotational symmetry is when an object is rotated around a center point (turned) a number of degrees and the object appears the same. The order of symmetry is the number of positions the object looks the same in a \(360^\circ\) rotation.
Steps:
Circle and square.
Chapter 14 Ex.14.3 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 rotational symmetry of order more than \(1\).
Solution
What is unknown?
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 rotational symmetry of order more than \(1\).
Reasoning:
To solve this question, remember the concept of rotational symmetry. Rotational symmetry is when an object is rotated around a centre point (turned) a number of degrees and the object appears the same. The order of symmetry is the number of positions the object looks the same in a \(360^\circ\) rotation.
Steps:
(i) An equilateral triangle has both line and rotational symmetry of order more than \(1.\)
(ii) An isosceles triangle has only one- line symmetry and no rotational symmetry of order more than \(1.\)
(iii) Parallelogram has two order of rotational symmetry but no line of symmetry.
(iv) A kite is a quadrilateral which has only one line of symmetry but not rotational symmetry of order more than \(1.\)
Chapter 14 Ex.14.3 Question 3
If a figure has two or more lines of symmetry, should it have rotational symmetry of order more than \(1\)?
Solution
What is unknown?
The figures have rotational symmetry of order more than \(1\).
Reasoning:
Rotational symmetry is when an object is rotated around a center point (turned) a number of degrees and the object appears the same. The order of symmetry is the number of positions the object looks the same in a \(360^\circ\) rotation.
Steps:
Yes, because every line through the centre forms a line of symmetry and it has rotational symmetry around the centre for every angle.
Chapter 14 Ex.14.3 Question 4
Fill in the blanks – Shape; Centre of Rotation; Order of Rotation; Angle of Rotation; for each of the figures respectively.
Shape |
Centre of Rotation |
Order of Rotation |
Angle of Rotation |
Square | |||
Rectangle | |||
Rhombus | |||
Equilateral Triangle | |||
Regular Hexagon | |||
Circle | |||
Semi-circle |
Solution
What is known?
Shapes
What is unknown?
Centre of rotation, order of rotation and angle of rotation.
Reasoning:
Centre of rotation: In case of polygon centre of rotation is intersecting point of diagonals and in case of circle centre of rotation is centre of the circle.
Order of rotation: In case of regular polygon order of rotation is always equals to the number of sides and in case of circle order of rotation is infinite.
Angle of rotation: In case of regular polygon angle of rotation is always equals to the measurement of one angle and in case of circle angle of rotation is at every point.
Steps:
Shape |
Centre of Rotation | Order of Rotation | Angle of Rotation |
Square | Intersecting points of diagonals | \(4\) | \(90^\circ\) |
Rectangle | Intersecting points of diagonals | \(2\) | \(180^\circ\) |
Rhombus | Intersecting points of diagonals | \(2\) | \(180^\circ\) |
Equilateral Triangle | Intersecting points of diagonals | \(3\) | \(120^\circ\) |
Regular Hexagon | Intersecting points of diagonals | \(6\) | \(60^\circ\) |
Circle | Mid- point of diameter | Infinite | At every point |
Semi-circle | Mid- point of diameter | \(1\) | \(360^\circ\) |
Chapter 14 Ex.14.3 Question 5
Name the quadrilaterals which have both line and rotational symmetry of order more than \(1\).
Solution
What is known?
Figures
What is unknown?
The quadrilaterals which have both line and rotational symmetry of order more than \(1\).
Reasoning:
To solve this question, you have to use the concept of line symmetry and rotational symmetry. The ‘Line of Symmetry’ is the imaginary line where you could fold the image and have both halves match exactly. Rotational symmetry is when an object is rotated around a centre point (turned) a number of degrees and the object appears the same. The order of symmetry is the number of positions the object looks the same in a \(360^\circ\) rotation.
Steps:
Square have both line and rotational symmetry of order more than \(1\).
Chapter 14 Ex.14.3 Question 6
After rotating by \( 60^\circ\) about a centre, a figure looks the same as its original position. At what other angles will this happen for the figure?
Solution
What is known?
Angle of rotation \(60^\circ\)
What is unknown?
At what other angles the figure looks the same as its original.
Reasoning:
If a figure looks same as the original on one angle then it will look same as original on all multiple of that angle.
Steps:
If figure looks the same as its original position by rotating at an angle of \(60^\circ\) it will also look same by rotating at angle of \(120^\circ, \;180^\circ,\; 240^\circ,\;300^\circ,\; 360^\circ\) as these are its multiples.
Chapter 14 Ex.14.3 Question 7
Can we have a rotational symmetry of order more than \(1\) whose angle of rotation is –
(i) \(45^\circ\)
(ii) \(17^\circ\)
Solution
What is known?
Angle of rotations
What is unknown?
The figures have rotational symmetry of order more than \(1\) with the angles \(45^\circ\) and \(17^\circ\)
Reasoning:
If the given angle is a factor of \(360^\circ,\) then the figure will have rotational symmetry of order more than one otherwise not.
Steps:
(i) \(45^\circ\) is a factor of \(360^\circ,\) so the figure will have a rotational symmetry of order more than \(1\) and there would be \(8\) rotations.
(ii) \(17^\circ\) is not a factor of \(360^\circ,\) so the figure will not have a rotational symmetry of order more than \(1.\)