Delving into the Art of Symmetry Chapter 14 of NCERT Class 7 Maths, "Symmetry," is your key to unlocking the world of geometric balance and harmony. This chapter's detailed solutions provide comprehensive answers to all questions from the NCERT textbook, helping students solidify their grasp on the subject. Expert educators have meticulously unraveled each problem, presenting a step-by-step approach with lucid explanations. With three distinct exercises within "Symmetry," our NCERT Solutions for Class 7 Math comprehensively tackle each exercise, offering a systematic path to problem resolution and exam preparedness. Exercise 14.1 is dedicated to the exploration of "Lines of Symmetry for a Regular Polygon." These solutions have been meticulously curated by subject matter experts to empower students in their exam readiness, ultimately supporting their quest to secure commendable marks in the realm of mathematics.
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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)
(a) 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.
The name of the figure is square.
(b) 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.
The name of the figure is triangle.
(c) 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.
The name of the figure is rhombus.
(d) 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.
The name of the figure is circle.
(e) 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.
The name of the figure is pentagon.
(f) 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.
The name of the figure is octagon.
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)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(a)
The figure given has 3 lines of symmetry.
So, it has multiple lines of symmetry.
(b)
The figure given has 2 lines of symmetry.
So, it has multiple lines of symmetry.
(c)
The figure given has 3 lines of symmetry.
So, it has multiple lines of symmetry.
(d)
The figure given has 2 lines of symmetry.
So, it has multiple lines of symmetry.
(e)
The figure given has 4 lines of symmetry.
So, it has multiple lines of symmetry.
(f)
The figure given has only 1 line of symmetry.
(g)
The figure given has 4 lines of symmetry.
So, it has multiple lines of symmetry.
(h)
The figure given has 6 lines of symmetry.
So, it has multiple lines of symmetry.
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?
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 in more than one way.
Copy the diagram and complete each shape to be symmetric about the mirror line(s):
(a)
(b)
(c)
(a)
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.
(b)
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.
(c)
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.
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
(a)
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.
An equilateral triangle which has 3 lines of symmetry is shown in the figure below,
(b)
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.
An isosceles triangle which has 1 line of symmetry is shown in the figure below,
(c)
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.
A scalene triangle which has no line of symmetry is shown in the figure below,
(d)
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.
A square which has 4 lines of symmetry is shown in the figure below,
(e)
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.
A rectangle which has 2 lines of symmetry is shown in the figure below,
(f)
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.
A rhombus which has 2 lines of symmetry is shown in the figure below,
(g)
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.
A parallelogram which has no line of symmetry is shown in the figure below,
(h)
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.
A quadrilateral which has no line of symmetry is shown in the figure below,
(i)
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.
A regular hexagon which has 6 lines of symmetry is shown in the figure below,
(j)
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.
A circle which has infinite lines of symmetry,
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
(a) a vertical mirror
The English letters which 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 letters having 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
Give three examples of shapes with no line of symmetry.
A shape has no line of symmetry if there is no line about which the figure may be folded, and parts of the figure will not coincide.
A scalene triangle, a quadrilateral and a parallelogram
What other name can you give to the line of symmetry of
(a) an isosceles triangle?
(b) a circle?
(a)
The other name for the line of symmetry of an isosceles triangle is median or altitude.
(b)
The other name for the line of symmetry of a circle is diameter.
Given the line(s) of symmetry, find the other hole(s):
(a)
(b)
(c)
(d)
(e)
(a) 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.
So, the other hole is shown in the figure below.
(b) 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.
So, the other hole is shown in the figure below.
(c) 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.
So, the other hole is shown in the figure below.
(d) 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.
So, the other hole is shown in the figure below.
(e) 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.
So, the other hole is shown in the figure below.
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)
(a) 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) 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) 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) 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.
(e) 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.
(f) 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.
(g) 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.
(h) 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.
(i) 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.
(j) 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.
(k) 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.
(l) 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.
Give the order of rotational symmetry for each figure:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(a)
The above figure has rotational symmetry of order 2.
(b)
The above figure has rotational symmetry of order 2.
(c)
The above figure has rotational symmetry of order 3.
(d)
The above figure has rotational symmetry of order 4.
(e)
The above figure has rotational symmetry of order 4.
(f)
The above figure has rotational symmetry of order 5.
(g)
The above figure has rotational symmetry of order 6.
(h)
The above figure has rotational symmetry of order 3.
Which of the following figures have rational symmetry of order more than 1:
(a)
So, the above figure has rotational symmetry of order 4.
(b)
So, the above figure has rotational symmetry of order 3.
(c) So, the given figure has rotational symmetry of order 1.
(d)
So, the above figure has rotational symmetry of order 2.
(e)
So, the above figure has rotational symmetry of order 3.
(f)
So, the above figure has rotational symmetry of order 4.
By observing all the figures (a), (b), (c), (d), (e) and (f) have rotational symmetry of order more than 1.
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?
The other angles are, 120°, 180°, 240°, 300°, 360°
So, the figure is said to have rotational symmetry about the same angle as the first one. Hence, the figure will look exactly the same when rotated by 60° from the last position.
Name any two figures that have both line symmetry and rotational symmetry.
Equilateral triangle and Circle.
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 have rotational symmetry of order more than 1.
Fill in the blanks:
Shape |
Centre of Rotation |
Order of Rotation |
Angle of Rotation |
Square |
|||
Rectangle |
|||
Rhombus |
|||
Equilateral Triangle |
|||
Regular Hexagon |
|||
Circle |
|||
Semi-circle |
Shape |
Centre of Rotation |
Order of Rotation |
Angle of Rotation |
Square |
Intersecting point of diagonals |
4 |
90o |
Rectangle |
Intersecting point of diagonals |
2 |
180o |
Rhombus |
Intersecting point of diagonals |
2 |
180o |
Equilateral Triangle |
Intersecting point of medians |
3 |
120o |
Regular Hexagon |
Intersecting point of diagonals |
6 |
60o |
Circle |
Centre |
Infinite |
Every angle |
Semi-circle |
Mid-point of diameter |
1 |
360o |
Name the quadrilaterals which have both line and rotational symmetry of order more than 1.
The quadrilateral, which has both line and rotational symmetry of order more than 1, is a square.
Line symmetry:
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) 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 is an isosceles triangle.
(iii) 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 is a rhombus.
Can we have a rotational symmetry of order more than 1 whose angle of rotation is
(i) 45°?
(ii) 17°?
(i) Yes. We can have a rotational symmetry of order more than 1 whose angle of rotation is 45o.
(ii) No. We cannot have a rotational symmetry of order more than 1 whose angle of rotation is 17o.
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