Class 6 Maths Chapter 9 Symmetry Notes
Class 6 Maths Chapter 9 Notes – Class 6 Symmetry Notes
In Mathematics, symmetry means a shape remains the same on Totaling or flipping. The concept of symmetry is commonly found in geometry. In symmetrical figures, some parts of the figure are repeated and these repetitions occur in a definite pattern. Symmetry is important in many fields, from art and architecture to nature and science. It helps create balance and harmony in design, and understanding it can make things more visually appealing and easier to understand. Some symmetrical designs are shown below:
Lines of Symmetry
A line that cuts a figure into two parts that exactly overlap when folded along that line is called a line of symmetry of the figure. A figure may have one or many lines of symmetry.
e.g. In the adjoining figure, l is called the line or axis of symmetry.
Figures with more than one line as symmetry.
The regular polygons are symmetrical figures. Each regular polygon has many lines of symmetry as it has many sides. For the regular polygons, multiple lines of symmetry are shown below.
Example 1.
Draw the lines of symmetry of an octagon.
Solution:
The lines of symmetry of an octagon are shown below.
Hence, an octagon has eight lines of symmetry.
Example 2.
Draw the line of symmetry of the given figures and find out several line symmetry.
Solution:
The figure has 2 lines of symmetry which are shown by dotted lines.
The figure has 4 lines of symmetry which are shown by dotted lines.
The figure has 3 lines of symmetry which are shown by dotted lines.
The given triangle is isosceles, thus this figure has 1 line of symmetry.
Example 3.
Give an example of a shape that has no line of symmetry.
Solution:
One such example of a shape that has no line of symmetry is a scalene triangle. It has all the sides with different measures.
Mirror Image/Reflection
When a figure is folded along a line of symmetry then the two parts overlap completely. It means the part of the figure on one side of the line of symmetry is reflected by the part of the figure on the other side of the line of symmetry. A figure having line lines of symmetry is also said to have reflection symmetry. Some examples of reflection are shown below.
Here, the dotted line is the line of symmetry. Also, observe that the shape is the same but its side has one side of the line of symmetry reversed.
Example 4.
In the following figures, the mirror line (i.e. the line or symmetry) is given as a dotted line. Complete each figure performing reflection in a 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 names of the figures you complete?
Solution:
On completing all the figures about their lines of symmetry which are given as a dotted line, we get the following figures:
Example 5.
Name the letters of the English alphabet having reflection symmetry (i.e. symmetry related to mirror reflection) about a vertical mirror.
Solution:
A shape has reflection symmetry, if one half of it is the mirror image of the other half. Letters of the English alphabet having reflection symmetry about a vertical mirror are A, H, I, M, O, T, U, V, W, X, and Y.
Example 6.
Choose the letters from the followings that has reflection symmetry about a horizontal mirror.
A, B, C, D, T, W, X
Solution:
A shape has reflection symmetry if one half of it is the mirror image of the other half. So, letters having reflection symmetry are B, C, D, and X.
Example 7.
From the numbers 2 to 9, find the number that has reflection symmetry about both the horizontal and vertical mirrors.
Solution:
The number having reflection symmetry about both the horizontal and vertical mirrors is 8.
Example 8.
Draw the reflection of the following letter concerning the given mirror line shown dotted.
Solution:
The reflection of the given figure is equidistant from the mirror line. (Note the changes in orientation)
Punching of Paper
In punching of paper for symmetrical design, the line of fold is the line of symmetry as shown below.
Example 9.
The figures given below are with punched holes, find the axes of symmetry and write the multiple lines of symmetry they have.
Solution:
It has one line of symmetry.
It has one line of symmetry.
It has one line of symmetry.
It has one line of symmetry.
Example 10.
Given the line(s) of symmetry, find the other hole(s).
Solution:
For the given line(s) of symmetry, the other hole(s) are marked in the given figures below.
Example Problems
Question 1.
Draw the line of symmetry in the following.
Answer:
Question 2.
Draw the lines of symmetry of the following figure.
Answer:
Question 3.
Find the number of lines of symmetry of the following.
Answer:
Four lines of symmetry.
Question 4.
What will be the mirror image of the following figures?
Answer:
Question 5.
In an Art class, students are asked to draw the mirror image of the word written on the board. What will be the correct image for the question?
Answer:
Question 6.
Find the axes of symmetry in the following figure.
Answer:
Question 7.
Find the other hole using symmetry in the following figure.
Answer:
Question 8.
Given the lines of symmetry, find the other holes.
Answer:
Question 9.
Follow the directions given below and form the final figure.
Answer:
Question 10.
Is it true that each polygon has as many lines of symmetry as it has sides? Give some examples and draw the figures.
Answer:
True, do it yourself.
Rotational Symmetry
If after a rotation, an object looks exactly the same, then we say that it has a rotational symmetry.
Centre of Rotation
The fixed point around which an object rotates is called the ‘center of Rotation’.
Order of Rotational Symmetry
In a complete turn (of 360°), the number of times an object looks the same is called the order of rotational symmetry.
e.g. The order of symmetry of a square is 4.
Angle of Rotational Symmetry
The angle through which the object rotates to look the same is called the angle of rotational Symmetry.
e.g. Rotational symmetry of a square
Let us rotate a square ABCD (given in Fig. (i)) to a full turn through four positions i.e. 90°, 180°, 270°, and 360° to attain the positions shown in Fig. (ii), Fig. (iii), Fig. (iv) and Fig. (v), respectively.
Each of the four times, the figure fits onto itself. So, it has rotational symmetry of order 4 about its center. In this case, we observe that
- The center of rotation is the center of the square.
- The angle of rotation is 90°.
- The direction of rotation is clockwise.
- The order of rotational symmetry is 4.
- When an object rotates, its shape and size do not change.
- An object rotates in two ways i.e. clockwise and anti-clockwise.
- Rotating a figure through 90° clockwise is the same as rotating it anti-clockwise through 270°.
- Rotating a figure through 180° clockwise is the same as rotating it anti-clockwise through 180°.
Example 1.
Explain the rotational symmetry of a rectangle.
Solution:
In a complete turn (i.e. 360°), the number of times the shape of an object looks the same as the original one is called the order of rotational symmetry.
Let us rotate a rectangle ABCD (as shown below) to a full turn through two positions i.e. 180° and 360° to attain the positions shown in fig (ii) and fig (iii), respectively.
Here, we can see that it has a rotational symmetry of order 2 and the center of rotation is the center of a rectangle.
Example 2.
What is the angle of rotation of the given figure?
Solution:
It is clear from the given figure that it can attain the same figure through 6 positions on each of the 6 positions it will overlap with the original figure.
There, the angle of rotation = \(\frac{360^{\circ}}{6}\) = 60°
Thus, the angle of rotation of the given figure is 60°.
Example 3.
What is the order of rotation of the given figure?
Solution:
The given figure will overlap with itself only if it rotates through 180°.
Since the angle of rotation is 180°.
∴ The order of rotation of the given figure = \(\frac{360^{\circ}}{180^{\circ}}\) = 2
Example 4.
Give the order of rotation symmetry of a figure about the point marked.
Solution:
The order of rotational symmetry of the given figure is 3.
All three positions are shown below.
Example 5.
Give the order of the rotational symmetry of the given figures about the point marked.
Solution:
The given figure will overlap at only one angle of rotational symmetry i.e. 360°.
Thus, the order of the rotational symmetry of the given figure is 1.
The given figure will overlap at two positions i.e. after rotation 180° and 360°.
Thus, the order of the rotational symmetry of the given figure is 2.
Example 6.
Give the order of rotational symmetry for each figure.
Solution:
The number of positions in which a figure can be rotated and still appear exactly as it was before the rotation, is called the order of symmetry.
- A star can be rotated 5 times along its tip and looks the same every time and the center of rotation will be the center of the star. Hence, its order of symmetry is 5.
- Around the point of rotation recycled logo attains 3 positions same as the original figure. Hence, the recycle logo has an order of symmetry 3.
- The Swastik symbol has an order of symmetry 4 above the point of rotation.
- The roundabout road sign has an order of symmetry of 3.
Symmetries of a Circle
When a circle is rotated about its center, it aligns perfectly with its original position, independent of the angle of rotation. This characteristic defines a circle’s rotational symmetry, where every angle of rotation results in the circle coinciding with itself. Additionally, any diameter of the circle serves as a line of reflection symmetry. When a point on the circumference of the circle is connected to the center, and the line is extended to form a diameter, this diameter divides the circle into two mirror-image halves. Thus, every diameter is a line of symmetry. These properties of rotational and reflection symmetry in a circle are also observed in many everyday objects, such as wheels, which display similar symmetrical characteristics.
Example 7.
If you rotate a circle by 90°, what will the circle look like?
Solution:
The circle will look the same, as it has rotational symmetry.
Example Problems
Question 1.
Show the required steps to rotate the given figure such that it fits onto itself.
Answer:
Question 2.
In the given figure, find the order of rotational symmetry.
Answer:
4
Question 3.
Determine the order of rotation of the following figure.
Answer:
1
Question 4.
Find the order of rotational symmetry of the following figures.
Answer:
(i) 1
(ii) 4
Question 5.
What is the resultant figure after rotating the given figure 270° clockwise?
Answer:
Question 6.
Determine the resultant figure after retaining the given figure at 240° anti-clockwise.
Answer:
The final figure after rotating at 240° is
Question 7.
Give an example of a figure having a line of symmetry but does not possess rotational symmetry.
Answer:
Isosceles triangle
Question 8.
How many degrees can you rotate a circle before it looks the same again?
Answer:
Infinite
Line Symmetry and Rotational Symmetry
We observe many shapes, which have both line symmetry and rotational symmetry. Some common figures having both lines of symmetry and rotational symmetry are listed in the table below:
Some more shapes are listed below, which have both lines of symmetry and rotational symmetry.
- The alphabetical letters H, I, and X have 2 lines of symmetry, while rotational symmetry is of order 2.
- The alphabet O has infinite lines of symmetry and rotational symmetry of infinite order.
Example 1.
The following figure shows both lines of symmetry and rotational symmetry.
Solution:
We can show lines of symmetry for a given figure as
In the above figure, the rotational angle will be 90°. So, rotational symmetry will take place as
The given figure will complete a full rotation in 4 steps. So, its order of rotation will be 4.
Example 2.
Complete the following table for the English alphabets S, H, O and E.
Solution:
We know that 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 and a figure has a rotational symmetry if, after a rotation, the figure looks exactly the same. Then, the complete table is shown below.
Example 3.
A circle is symmetrical about each one of its diameters as shown in the following figure and therefore it has an unlimited number of lines of symmetry. Does a semi-circle also have an unlimited number of lines of symmetry?
Solution:
Since a semi-circle has only one diameter.
∴ A semi-circle AOBC has one line of symmetry, namely the perpendicular bisector (l) of its diameter AB as shown below.
Hence, the semi-circle doesn’t have an unlimited number of lines of symmetry.
Example 4.
The given figure is a rhombus. It is symmetrical about each one of its diagonals i.e. there are two lines of symmetry for rhombus. How many lines of symmetry can there be in a kite PQRS?
Solution:
The following figure represents a kite PQRS.
Since, in the kite (shown in the figure)
QR = RS and PQ = PS
It is symmetrical about its diagonal PR.
Thus, it has only one line of symmetry.
Example 5.
A regular hexagon is cut along one of its lines of symmetry as shown.
Is it true that the angle of rotation of the new figure is half of the previous figure? If not, at what angle it should be turned to get back to the original position?
Solution:
No, it’s not true.
The angle of rotation of the half-figure would be 360°. That is, it has no rotational symmetry.
Example 6.
If a triangle has both line and rotational symmetries of order more than 1. Name the triangle first and draw a rough sketch for the given condition.
Solution:
The triangle is equilateral. An equilateral triangle has 3 lines of symmetry which are shown by dotted lines.
Also, an equilateral triangle has rotational symmetry of order 3 as shown below.
Example 7.
Show by drawing, if a triangle has only a line of symmetry and no rotational symmetry of order more than 1.
Solution:
An isosceles triangle has only one line of symmetry, which is shown below by dotted lines and it has no rotational symmetry of order more than 1.
Example 8.
A quadrilateral with a rotational symmetry of order more than 1, but not a line symmetry is possible? Draw a rough figure.
Solution:
A quadrilateral i.e. parallelogram has rotational symmetry of order more than 1, but not a line symmetry.
Example 9.
Draw a rough figure for a quadrilateral with a line of symmetry, but not a rotational symmetry of order more than 1.
Solution:
Isosceles trapezium has a line of symmetry, which is shown by dotted lines, but not a rotational symmetry of order more than 1.
Example 10.
Is it a true statement that if a figure has two or more lines of symmetry, it should have rotational symmetry of order more than 1?
Solution:
Yes, the statement is true.
For example, a square has four lines of symmetry and rotational symmetry of order 4.
Example 11.
After rotating by 120° about a center, a figure looks the same as its original position. At what other angles will this happen for the figure?
Solution:
After rotation by 120° about a center, a figure looks the same as its original position. This will also happen for the figure at angles 240° and 360°, respectively.
Example 12.
Can we have a rotational symmetry of order more than 1 whose angle of rotation is
(i) 60°
(ii) 19°
Solution:
We know that an angle of rotation in one complete turn is 360°. So, if the given angle of rotation divides 360° completely, then for rotational symmetry of order more than 1 that angle will be possible, otherwise not possible.
(i) Yes, we can have a rotational symmetry of order more than 1 whose angle of rotation is 60°.
[∵ 360° is divisible by 60° completely]
(ii) No, we cannot have a rotational symmetry of order more than 1 whose angle of rotation is 19°.
[∵ 360° is not completely divisible by 19°]
Example Problems
Question 1.
The following figure shows both lines of symmetry and rotational symmetry.
Answer:
Do yourself.
Question 2.
Determine the line of symmetry of the following figure, if any.
Answer:
Line of Symmetry = 1
Question 3.
For the given figure, draw the line of symmetry and find the order of rotational symmetry.
Answer:
Line of symmetry (horizontal), order of rotational symmetry is 1.
Question 4.
Determine the line of symmetry and order of rotational symmetry, if any of the following figure.
Answer:
Line of symmetry = 4
Order of rotational symmetry = 4 (rotating at 90°).
Question 5.
Complete the table for the English alphabets Z, D, N, and C.
Answer:
Question 6.
List any three letters, which have some lines of symmetry and the order of rotational symmetry equal to 2.
Answer:
H, I, X have 2 lines of symmetry and the order of rotational symmetry is also 2.
Question 7.
Joseph is asked to cut the given figure in such a way that it focuses on two similar images. Determine how he will cut the figure accordingly.
Also, find the order of rotational symmetry of the above figure.
Answer:
The order of rotational symmetry is 1.
Question 8.
In the given figure, find the line of symmetry and rotational symmetry angle.
Answer:
(i) Line of symmetry = 4 and angle of rotation = 90°
(ii) Line of symmetry = 3 and angle of rotation = 120°
(iii) Do not show line of symmetry but show rotational symmetry, so angle of rotation = 90°
Question 9.
Give an example of a figure that has three lines of symmetry and rotational symmetry of order more than 1.
Answer:
An equilateral triangle.
Question 10.
After rotating by 30° about a center, a figure looks the same as its original position. At what other angles will this happen for the figure?
Answer:
60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, 300°, 330°, 360°
Question 11.
Can we have a rotational symmetry of order more than 1 whose angle of rotation is
(i) 40°
(ii) 23°
Answer:
(i) Yes
(ii) No
Question 12.
Give two examples of figures that have both lines of symmetry and rotational symmetry.
Answer:
Square and Circle
Question 13.
Name any four figures, that have the same number of lines of symmetry and order of rotational symmetry.
Answer:
Square, rectangle, equilateral triangle, and rhombus have the same number of lines of symmetry and rotational symmetry.
Question 14.
Fill in the blanks.
(i) An equilateral triangle and a circle have both _________ symmetry and _________ symmetry.
Answer:
An equilateral triangle and a circle have both line symmetry and rotational symmetry.
(ii) Equilateral triangles have both line and rotational symmetries of order more than _________
Answer:
Equilateral triangles have both line and rotational symmetries of order more than 1.
(iii) An isosceles triangle has only _________ line of symmetry.
Answer:
An isosceles triangle has only one line of symmetry.
(iv) A parallelogram has _________ symmetry of order more than 1, but not a _________ symmetry.
Answer:
A parallelogram has rotational symmetry of order more than 1, but not a line symmetry.
→ There are many figures or objects which when folded along a straight line are divided equally into two identical halves such that one part coincides with the other. Such figures are called symmetrical figures.
→ The line that divides a symmetrical figure into two equal parts that exactly overlap when folded along that line is called the line of symmetry. The idea of symmetry is used in almost all activities of our day-to-day life.
→ The concept of symmetry is exhibited in nature such as in beehives, tree leaves, flowers, etc. Symmetrical designs are used by artists, professionals, designers of jewelry or clothing, car manufacturers and building architects, etc.
→ A symmetrical figure may have one or more than one line of symmetry. Also, a figure or an object that has a line (s) of symmetry is said to have reflection symmetry.
Note:
1. Symmetry means the exact match in size and shape between two halves, parts, or sides of a figure or an object.
2. The line of symmetry can be real or imaginary.
3. Symmetrical figures having more than two lines of symmetry are said to be having multiple lines of symmetry. Sometimes a figure looks the same when it is rotated by an angle about a fixed point. Such figures or objects are said to have rotational symmetry.
Such an angle is called the angle of rotational symmetry of the figure. The fixed point of the figure about which the rotation occurs is called the center of rotation.
Note: If a figure comes back to its original shape after one complete rotation through 360°, then the figure does not have rotational symmetry.
→ Both halves of a symmetrical shape match exactly when folded on the line of symmetry or axis of symmetry.
→ The line of symmetry acts as the mirror.
→ The mirror reflection and the line of symmetry are related to each other.
→ An isosceles triangle has only one line of symmetry.
→ A scalene triangle has no line of symmetry.
→ Regular polygons have equal sides and equal angles. They have more than one (multiple) line of symmetry.
→ An equilateral triangle has three 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.
→ A circle has countless lines of symmetry.
→ The object and its image are symmetrical about the mirror-line.
→ In a kaleidoscope usually two mirrors forming a V-shape are used. The angle between the mirrors determines the number of lines of symmetry.
→ If, after a rotation, an object looks the same, it is said to have a rotational symmetry.
→ Some figures may have both lines of symmetry as well as angles of symmetry.
→ A quarter turn means 90°. A half-turn means a rotation by 180°. A three-quarter turn means 270°, and a full turn means 360°.
CBSE Class 6 Maths Chapter 9 Notes Symmetry
1. The objects or shapes with evenly balanced proportions are called symmetrical objects or shapes.
2. When a figure can be folded into two halves and both halves overlap each other, the figure is said to be symmetrical.
3. The line along which the figure is folded is called line of symmetry.
4. The line of symmetry can be horizontal, vertical, or slant.