NCERT Solutions for Class 7 Maths Chapter 14 - Symmetry

Exercise 14.1

1. Copy the figures with punched holes and find the axes of symmetry for the following:

a) 

Ans: In the above question ABCD is a symmetrical square and E and F are two holes.

The line that divides symmetrically the two holes is given by GH as shown

b) 

Ans: In the above question ABCD is a symmetrical square and E and F are two holes.

The line that divides symmetrically the two holes is given by BD as shown

c) 

Ans: In the above question ABCD is a symmetrical square and E and F are two holes.

The line that divides symmetrically the two holes is given by GH as shown

d) 

Ans: In the above question ABCD is a symmetrical square and E, F, G and H  are holes.

The line that divides symmetrically the holes is given by IJ as shown

e) 

Ans: In the above question ABCD is a symmetrical square and E, F, G and H  are holes.

There are four lines that divides symmetrically the two holes, given by AC,BD,IJ,KL as shown

f) 

Ans: In the above question ABCD is symmetrical square and E, F and H are three holes. F and H being overlapped with D and C respectively.

There is only one line that divides symmetrically the three holes, given by IJ as shown

g) 

Ans: In the above question ABD is a symmetrical triangle and E and D are two holes.

There is only one line that divides symmetrically the two holes, given by AF as shown

h) 

Ans: In the above question ABC is a symmetrical triangle and E and D are two holes.

There is only one line that divides symmetrically the two holes, given by AF as shown

i) 

Ans: In the above question ABC is a symmetrical triangle and E and D are two holes.

There is only one line that divides symmetrically the two holes and the triangle, given by AF as shown

j) 

Ans: Given we have a circle centred at E where D and C are two holes and hence the lines of symmetry EF,IJ,KL,GH are given by as shown

k) 

Ans: Given we have a circle where D,E and F,G are four holes and hence the lines of symmetry IJ, HG, KL, MN are given by as shown

l) 

Ans: Given we have a circle where D, E and C are three holes and hence the lines of symmetry GF is given as shown

2. Given the line(s) of symmetry, find the other hole(s):

a) 

Ans: Given AC is a line of symmetry and E is a hole 

Hence the other hole ${{E}_{1}}$ such that AC divides E and${{E}_{1}}$symmetrically as given by

b) 

Ans: Given FG is a line of symmetry and E is a hole 

Hence the other hole ${{E}_{1}}$ such that AC divides E and${{E}_{1}}$symmetrically as given by

c) 

Ans: Given AD is a line of symmetry and E is a hole 

Hence the other hole ${{E}_{1}}$ such that AC divides E and${{E}_{1}}$symmetrically as given by

d) 

Ans: Given we have DC is the line of symmetry and E is the one of the holes which is divided symmetrically by the line DC.

Hence the second hole ${{E}_{1}}$ is given as shown

e) 

Ans: Given we have DC is the line of symmetry and ${{E}_{1}}$ and E are holes which are divided symmetrically by the line DC.

Hence the other holes ${{E}_{2}}$ and ${{E}_{3}}$ are given as shown

3. In the following figures, the mirror line DC(i.e., the line of symmetry) is given. Complete each figure performing reflection in (mirror) line. (You might perhaps place a mirror along the line DC and look into the mirror for the image). Are you able to recall the name of the figure you completed?

a) 

Ans: Given we have line of symmetry is DC and hence after seeing the image on the mirror we get the complete figure of square as shown

b) 

Ans: Given we have a line of symmetry is CD and hence after seeing the image on the mirror we get the complete figure of triangle as shown 

c) 

Ans: Given we have a line of symmetry is DC and hence after seeing the image on the mirror we get the complete figure of rhombus as shown 

d) 

Ans: Given we have a line of symmetry is CD and hence after seeing the image on the mirror we get the complete figure of circle as shown 

e) 

Ans: Given we have a line of symmetry is DC and hence after seeing the image on the mirror we get the complete figure of pentagon as shown 

f) 

Ans: Given we have a line of symmetry is DC and hence after seeing the image on the mirror we get the complete figure of octagon as shown 

4. 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) 

Ans: Given we have a rectangle and the figure with multiple lines of symmetry is given as shown

b) 

Ans: Given we have a square and the figure with multiple lines of symmetry is given as shown

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?

Ans: From the give figure we have AB,CD,EF,GH as lines of symmetry hence it has more than one line of symmetry

Also the more shaded part about any one diagonal is as shown

6. Copy the diagram and complete each shape to be symmetric about the mirror line(s):

a) 

Ans: Given the figure in which CE is the line of symmetry 

The complete figure is as shown

b) 

Ans: Given the figure in which BF, GH is the lines of symmetry 

The complete figure is as shown

c) 

Ans: Given the figure in which BF, GH is the lines of symmetry 

The complete figure is as shown

d) 

Ans: Given the figure in which BF, GH is the lines of symmetry 

The complete figure is as shown

7. State the number of lines of symmetry for the following figures:

(a) An equilateral triangle

Ans: An equilateral triangle has $3$lines of symmetry AD, BE, CF as shown

(b) An isosceles triangle 

Ans: An isosceles triangle has $1$line of symmetry as shown

(c) A scalene triangle

Ans: A scalene triangle has no lines of symmetry as shown

(d) A square 

Ans: A square has $4$lines of symmetry CD, DF, HI, JK as shown

(e) A rectangle 

Ans: A rectangle has four lines of symmetry CE, DF, HI, JK as shown

(f) A rhombus

Ans: A rhombus has $2$lines of symmetry AC, BD as shown

(g) A parallelogram 

Ans: A parallelogram has no lines of symmetry

(h) A quadrilateral 

Ans: A quadrilateral has no lines of symmetry

(i) A regular hexagon

Ans: A regular hexagon has $6$lines of symmetry AD, KL, FC, HG, BE, IJ, as shown

(j) A circle

Ans: A circle has infinite lines of symmetry, IJ, KL, GH, MN and many more.

8. at letters of the English alphabet have reflectional symmetry (i.e., symmetry related to mirror reflection) about:

(a) a vertical mirror

Ans: It can easily be observed that A, H, I, M, O, T, U, X, V, W, Y are the letters which show symmetry about the vertical line

(b) a horizontal mirror

Ans: It can easily be observed that B, C, D, E, H, I, O, X are the letters which show symmetry about the horizontal line

(c) both horizontal and vertical mirrors

Ans: It can easily be observed that H, I, O, X are the letters which show symmetry both about the vertical line and the horizontal line.

9. Give three examples of shapes with no line of symmetry.

Ans: The three shapes are written as

1. An scalene triangle

2. Any general quadrilateral

3. Any general parallelogram

10. What other name can you give to the line of symmetry of:

(a) an isosceles triangle?

Ans: An isosceles triangle  has altitude and median as line of symmetry 

(b) a circle?

Ans: A circle has any diameter as line of symmetry

Exercise 14.2

1. Which of the following figures have rotational symmetry of order more than 1:

(a)  

Ans: The above figure has rotational symmetry of order more than one

(b) 

Ans: The above figure has rotational symmetry of order more than one

(c) 

Ans: The above figure has rotational symmetry of order not more than one

(d) 

Ans: The above figure has rotational symmetry of order more than one

(e)

Ans: The above figure has rotational symmetry of order more than one

(f) 

2. Give the order the rotational symmetry for each figure:

(a) 

Ans: The above figure has rotational symmetry of order is $2$, when turn of ${{180}^{\circ }}$takes place

(b) 

Ans: The above figure has rotational symmetry of order is $2$, when turn of ${{180}^{\circ }}$takes place

(c) 

Ans: The above figure has rotational symmetry of order is $3$, when turn of ${{120}^{\circ }}$takes place

(d) 

Ans: The above figure has rotational symmetry of order is $4$, when turn of ${{90}^{\circ }}$takes place

(e) 

Ans: The above figure has rotational symmetry of order is $4$, when turn of ${{90}^{\circ }}$takes place

(f) 

Ans: The above figure has rotational symmetry of order is $5$, when turn of ${{72}^{\circ }}$takes place

(g) 

Ans: The above figure has rotational symmetry of order is $6$, when turn of ${{60}^{\circ }}$takes place

(h) 

Ans: The above figure has rotational symmetry of order is $3$, when turn of ${{120}^{\circ }}$takes place

Exercise 14.3

1. Name any two figures that have both line symmetry and rotational symmetry.

Ans: A circle and an equilateral triangle have both line and rotational symmetry

2. Draw, wherever possible, a rough sketch of:

(i). a triangle with both line and rotational symmetries of order more than 1.

Ans: An equilateral triangle is the triangle that contains line symmetry as shown

And rotational symmetry of ${{120}^{\circ }}$as shown

(ii). a triangle with only line symmetry and no rotational symmetry of order more than 1.

Ans: An isosceles triangle is the triangle that contains only one line symmetry and no rotational symmetry of order more than one as shown

(iii). a quadrilateral with a rotational symmetry of order more than 1 but not a line symmetry.

Ans: We can’t think of such quadrilaterals such that it has order of rotational symmetry more than one but no line symmetry

(iv). a quadrilateral with line symmetry but not a rotational symmetry of order more than 1.

Ans: In this case we can think of a trapezium with equal non parallel sides have a line symmetry but not any rotational symmetry of order more than one as shown

3. In a figure has two or more lines of symmetry, should it have rotational symmetry of order more than 1?

Ans: If those lines of symmetry passes through the centre of the figure then surely it should have rotational symmetry of order more than one.

4. Fill in the blanks:

Ans: The above shown table can be filled as shown

5. Name the quadrilateral which has both line and rotational symmetry of order more than 1.

Ans: A square has both line and rotational symmetry of order more than one

6. After rotating by${{60}^{\circ }}$about a centre, a figure looks exactly the same as its original position. At what other angles will this happen for the figure?

Ans: Other angles will be ${{120}^{\circ }}$, ${{180}^{\circ }}$,${{360}^{\circ }}$

For ${{120}^{\circ }}$there will be three rotations as shown

For ${{180}^{\circ }}$there will be two rotations as shown

For ${{360}^{\circ }}$there will be one rotations as shown

Can we have a rotational symmetry of order more than one whose angle of rotation is:

(i). ${{45}^{\circ }}$

Ans: If the angle is \[{{45}^{\circ }}\]then symmetry is possible because ${{360}^{\circ }}$ is divisible by \[{{45}^{\circ }}\]and hence we will get \[8\]rotations

(ii). ${{17}^{\circ }}$

Ans: If the angle is \[{{17}^{\circ }}\]then symmetry is not possible because ${{360}^{\circ }}$ is not divisible by \[{{45}^{\circ }}\]

NCERT Solutions for Class 7 Maths Chapter 14 Free PDF Download

What is Symmetry?

Symmetry is an important geometrical concept in mathematics. The idea of symmetry is used in almost all activities of our day-to-day life. When any object is divided into two identical halves then they form a mirror image of each other. The two identical halves are called symmetrical to each other. 

Facts

• Regular polygons have equal sides and equal angles. They have more than one or multiple lines of symmetry.

• A square has four lines of symmetry.

• A regular pentagon has five lines of symmetry.

• A regular hexagon has six lines of symmetry.

• Rotation turns an object about a fixed point that is called the center of rotation.

• The angle by which an object rotates is called the angle of rotation.

• A half-turn means a rotation by 1800.

• After a rotation, if an object looks exactly the same, it is said to exhibit rotational symmetry.

Line of Symmetry

In this section of chapter 14 for class 7, you will learn the meaning of the line of symmetry for a figure. When a line divides a shape or a figure into two identical halves, we say that the shape exhibits symmetry. The line that divides the shape is called the line of symmetry. A figure has a line of symmetry, if there is a line about which the figure can be folded into two coinciding parts. The line of symmetry is also called the axis of symmetry. 

The concept of symmetry is exhibited in nature in beehives, tree leaves, flowers, etc. Symmetrical designs are used by artists, manufacturers, designers, architects, etc. 

Line of Symmetry for Regular Polygons

We know that a polygon is a closed figure made of only line segments. The polygon that is made up of the least number of line segments is a triangle. A polygon is said to be regular if all the sides are of equal length and all its angles are of equal measure. An equilateral triangle is a regular polygon. A square is also a regular polygon.

An equilateral triangle is a regular polygon because each of its sides has the same length and each of its angles measures 600. An equilateral triangle has three lines of symmetry as shown in the figure below.

(Image will be uploaded soon)

A square is also a regular polygon because all its sides are equal and each of its angles is a right angle. 

A square has four lines of symmetry.

Each side of a regular polygon is of equal length and each of its angles measures 1080. A pentagon has five lines of symmetry. 

A regular hexagon has all its sides equal and each of its angles measures 1200. A regular hexagon has six lines of symmetry, each line through each of its vertices. 

Note: A shape is said to have a line of symmetry when one-half of it is the mirror image of the other. So, the concept of line symmetry is closely related to mirror reflection.

Rotational Symmetry

In this part of chapter 14 for class 7, you will learn about Rotational Symmetry. This will provide you with an explanation of the NCERT solutions. 

The shape and size of an object do not change when it rotates. The rotation takes place about a fixed point. This fixed point where the rotation occurs is called the center of rotation. The angle through which the body makes turns is called the angle of rotation.

1. A full-turn means a rotation by 3600. 

2. A half-turn means a rotation by 1800. 

3. A quarter-turn means a rotation by 900. 

If a figure can be rotated less than 3600 around a point such that it coincides with itself, it has rotational symmetry. 

Line of Symmetry and Rotational Symmetry

There are some shapes, which have both line symmetry and rotational symmetry. 

Example:    

(i) A square has line symmetry as well as rotational symmetry.

(ii) An equilateral triangle has line symmetry as well as rotational symmetry.

(iii) A regular polygon has a line symmetry as well as rotational symmetry. 

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Delve into the world of symmetry with Vedantu's NCERT Solutions for Class 7 Maths Chapter 14. Designed to enhance conceptual understanding, these solutions offer comprehensive insights into symmetry-related topics. Explore the benefits of referring to Vedantu's NCERT Solutions for a holistic grasp of Class 7 Maths concepts. Some of the features of Vedantu’s NCERT Solutions for Class 7 Maths Chapter 14 - Symmetry include:

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Conclusion

Symmetry is an important topic from the examination point of view. If you have a good understanding of the concepts of chapter 14, Symmetry then you can easily score good marks in it. You will also acquire a good knowledge of symmetry and can use it in your daily activities. The concepts and the key features for the NCERT Solutions to chapter 14 for class 7 are designed by subject matter experts. You can download the free PDF from Vedantu and you can also get in touch with the experts for further clarification anytime.