Class 6 Maths Chapter 9 Notes Symmetry
Class 6 Maths Notes Chapter 9 – Class 6 Symmetry Notes
→ When a figure is made up of parts that repeat in a definite pattern, we say that the figure has symmetry. We say that such a figure is symmetrical.
→ A line that cuts a plane figure into two parts that exactly overlap when folded along that line is called a line of symmetry or axis of symmetry of the figure.
→ A figure may have multiple lines of symmetry.
→ Sometimes a figure looks the same when it is rotated by an angle about a fixed point. Such an angle is called an angle of symmetry of the figure. A figure that has an angle of symmetry strictly between 0 and 360 degrees is said to have rotational symmetry. The point of the figure about which the rotation occurs is called the center of rotation.
→ A figure may have multiple angles of symmetry.
→ Some figures may have a line of symmetry but no angle of symmetry, while others may have angles of symmetry but no lines of symmetry. Some figures may have both lines of symmetry as well as angles of symmetry.
Look around you — you may find many objects that catch your attention. Some such things are shown below:
There is something beautiful about the pictures above. The flower looks the same from many different angles. What about the butterfly? No doubt, the colours are very attractive. But what else about the butterfly appeals to you?
In these pictures, it appears that some parts of the figure are repeated and these repetitions seem to occur in a definite pattern. Can you see what repeats in the beautiful rangoli figure? In the rangoli, the red petals come back onto themselves when the flower is rotated by 90˚ around the center and so do the other parts of the rangoli.
Now, can you say what figure repeats along each side of the hexagon? What is the shape of the figure that is stuck to each side? Do you recognize it? How do these shapes move as you move along the boundary of the hexagon? What about the other pictures — what is it about those structures that appeals to you and what are the patterns in those structures that repeat?
On the other hand, look at this picture of clouds. There is no such repetitive pattern. We can say that the first four figures are symmetrical. and the last one is not symmetrical. Symmetry refers to a part or parts of a figure that are repeated in some definite pattern.
What are the symmetries that you see in these beautiful structures?
Line of Symmetry Class 6 Notes
→ If one half of a shape can be folded into the other half, that is one half of the figure would fit exactly into the other half, then the figure is said to have symmetry.
→ If a line divides a figure into two parts such that when the figure is folded about this line, the two parts of the figure coincide, then this line is known as the line of symmetry or axis of symmetry.
→ A figure having more than one line of symmetry ¡s said to have multiple lines of symmetry.
Figures with more than one line of symmetry
→ A figure that has a line(s) of symmetry is thus also said to have reflection symmetry.
→ Symmetrical shapes can be created by paper folding and cutting or punching holes.
Figure (a) shows the picture of a blue triangle with a dotted line. What if you fold the triangle along the dotted line? Yes, one half of the triangle covers the other half completely. These are called mirror halves!
What about Figure (b) with the four puzzle pieces and a dotted line passing through the middle? Are they mirror halves? No, when we fold along the line, the left half does not exactly fit over the right half. 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.
Does a square have only one line of symmetry?
Take a square piece of paper. By folding, find all its lines of symmetry.
Here are the different folds giving different lines of symmetry.
- Fold the paper in half vertically.
- Fold it again into half horizontally. (i.e. you have folded it twice).
- Now open out the folds.
Again fold the square into half (for a third time now), but this time along a diagonal, as shown in the figure. Again, open it.
Fold it into half (for the fourth time), but this time along the other diagonal, as shown in the figure. Open out the fold.
Is there any other way to fold the square so that the two halves overlap? How many lines of symmetry does the square shape have?
Thus, figures can have multiple lines of symmetry. The figures below also have multiple lines of symmetry. Can you find them all?
We saw that the diagonal of a square is also a line of symmetry. Let us take a rectangle that is not a square. Is its diagonal a line of symmetry?
First, see the rectangle and answer this question. Then, take a rectangular piece of paper and check if the two parts overlap by folding it along its diagonal. What do you observe?
Reflection
So far we have been saying that when we fold a figure along a line of symmetry, the two parts overlap completely. We could also say that the part of the figure on one side of the line of symmetry is reflected by the line to the other side; similarly, the part of the figure on the other side of the line of symmetry is reflected to the first side! Let us understand this by labeling some points on the figure.
The figure shows a square with its corners labeled A, B, C, and D. Let us first consider the vertical line of symmetry. When we reflect the square along this line, points B, and C on the right get reflected to the left side and occupy the positions occupied earlier by A, and D. What happens to points A, and D? A occupies the position occupied by B and D that of C!
What if we reflect along the diagonal from A to C? Where do points A, B, C, and D go? What if we reflect along the horizontal line of symmetry? A figure that has a line or lines of symmetry is thus also said to have reflection symmetry.
Generating Shapes Having Lines of Symmetry
So far we have seen symmetrical figures and asymmetrical figures. How does one generate such symmetrical figures? Let us explore this.
Ink Blot Devils
You enjoyed doing this earlier in Class 5. Take a piece of paper. Fold it in half. Open the paper and spill a few drops of ink (or paint) on one half. Now press the halves together and then open the paper again.
- What do you see?
- Is the resulting figure symmetric?
- If yes, where is the line of symmetry?
- Is there any other line along which it can be folded to produce two identical parts?
- Try making more such patterns.
Paper Folding and Cutting
Here is another way of making symmetric shapes! In these two figures, a sheet of paper is folded and a cut is made along the dotted line shown. Sketch how the paper will look when unfolded.
Do you see a line of symmetry in this figure? What is it?
Make different symmetric shapes by folding and cutting. There are more ways of folding and cutting pieces of paper to get symmetric shapes! Use thin rectangular coloured paper. Fold it several times and create some intricate patterns by cutting the paper, like the one shown here. Identify the lines of symmetry in the repeating design. Use such decorative paper cut-outs for festive occasions.
Rotational Symmetry Class 6 Notes
→ Sometimes a figure looks the same when it is rotated by an angle about a fixed point. This fixed point is called the center of rotation. Such a figure is said to have rotational symmetry.
→ The angle by which the figure is rotated is called the angle of rotational symmetry or just the angle of symmetry. This angle lies between 0 and 360 degrees.
→ A figure may have multiple angles of symmetry.
→ Smallest angle of symmetry = 360 ÷ (number of lines of symmetry in the figure)
→ Other angles are multiples of the smallest angle till we reach 360°.
→ Also, number of angles of symmetry = number of lines of symmetry.
→ The number of angles of symmetry is also called the order of rotational symmetry.
→ Minimum order of rotation symmetry is 1 as any object will look like itself after a complete turn.
The paper windmill in the picture looks symmetrical but there is no line of symmetry! However you fold it, the two halves will not exactly overlap. On the other hand, if you rotate it by 90° about the red point at the center, the windmill looks the same. We say that the windmill has rotational symmetry.
When talking of rotational symmetry, there is always a fixed point about which the object is rotated. This fixed point is called the center of rotation. Will the windmill above look the same when rotated through an angle of less than 90°? No!
An angle through which a figure can be rotated to look the same is called an angle of rotational symmetry, or just an angle of symmetry, for short. For the windmill, the angles of symmetry are 90° (quarter turn), 180° (half turn), 270° (three-quarter turn) and 360° (full turn). Observe that when any figure is rotated by 360°, it comes back to its original position, so 360° is always an angle of symmetry. Thus, we see that the windmill has 4 angles of symmetry.
Do you know of any other shape that has exactly four angles of symmetry? How many angles of symmetry does a square have? How much rotation does it require to get the initial square?
We get back a square overlapping with itself after 90° of rotation. This takes point A to the position of point B, point B to the position of point C, point C to the position of point D, and point D back to the position of point A. Do you know where to mark the center of rotation?
What are the other angles of symmetry?
Example:
Find the angles of symmetry of the following strip.
Solution:
Let us rotate the strip in a clockwise direction about its center.
A rotation of 180° results in the figure above. Does this overlap with the original figure?
No. Why? Another rotation through 180° from this position gives the original shape. This figure comes back to its original shape only after one complete rotation through 360°. So we say that this figure does not have rotational symmetry.
Rotational Symmetry of Figures with Radial Arms
Consider this figure, a picture with 4 radial arms. How many angles of symmetry does it have? What are they? Note that the angle between adjacent central dotted lines is 90°. Can you change the angles between the radial arms so that the figure still has 4 angles of symmetry? Try drawing it.
To check if the figure is drawn indeed has 4 angles of symmetry, you could draw the figure on two different pieces of paper. Cut out the radial arms from one of the papers. Keep the figure on the paper fixed and rotate the cutout to check for rotational symmetry. How will you modify the figure above so that it has only two angles of symmetry? Here is one way:
We have seen figures having 4 and 2 angles of symmetry. Can we get a figure having exactly 3 angles of symmetry? Can you use radial arms for this? Let us try with 3 radial arms as in the figure below. How many angles of symmetry does it have and what are they? Here is a figure with three radial arms.
Trace and cut out a copy of this figure. By rotating the cutout over this figure determine its angles of rotation. We see that only a full turn or a rotation of 360° will bring the figure back into itself. So this figure does not have rotational symmetry as 360 degrees is its only angle of symmetry. However, can anything in the figure be changed to make it have 3 angles of symmetry?
Can it be done by changing the angles between the dotted lines? If a figure with three radial arms should have rotational symmetry, then a rotated version of it should overlap with the original. Here are rough diagrams of both of them. If these two figures must overlap, what can you tell about the angles?
Observe that ∠A must overlap ∠B, ∠B must overlap ∠C and ∠C must overlap ∠A. So, ∠A = ∠B = ∠C. What must this angle be?
We know that a full turn has 360 degrees. This is equally distributed amongst these three angles. So each angle must be 360°/3 = 120°. So, the radial arms figure with 3 arms shows rotational symmetry when the angle between the adjacent dotted lines is 120 deg. Use paper cutouts to verify this observation. Now how many angles of rotation does the figure have and what are they?
Note: The colours have been added to show the rotations.
Symmetries of a Circle
The circle is a fascinating figure. What happens when you rotate a circle clockwise about its center? It coincides with itself. It does not matter what angle you rotate it by! So, for a circle, every angle is an angle of symmetry.
Now take a point on the rim of the circle and join it to the center. Extend the segment to the diameter of the circle. Is that diameter a line of reflection symmetry? It is. Every diameter is a line of symmetry! Like wheels, we can find other objects around us having rotational symmetry. Find them. Some of them are shown below:
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In our daily life symmetry is a common term. When we see a figure with evenly balanced proportions, then we say that it is symmetrical.
If we can fold a picture in half such that the left and right halves match exactly, then the picture is said to have line symmetry. We can see that the two halves are mirror images of each other. If we place a mirror on the fold, then the image of one side of the picture will fall exactly on the other side of the picture. The line of the fold is called the line of symmetry. It divides the figure into two identical parts.
Making Symmetric Figures: Ink-Blot Devils
We can list a few objects from our surroundings of symmetry for these symmetric objects. Also, we can identify the lines
Figures with Two Lines of Symmetry
If we take a rectangular sheet and fold it length-wise or breadth-wise, we find that one half fits exactly over the other half. We say that a rectangle has two lines of symmetry.
Note: An isosceles triangle has only one line of symmetry.
A scalene triangle has no line of symmetry.
Figures with Multiple (Morethan Two) Lines of Symmetry
An Equilateral Triangle has three lines of symmetry whereas a circle has countless lines of symmetry.
Reflection and Symmetry
The line symmetry is closely related to mirror reflection. In mirror reflection, we have to take into account the left ↔ right changes in orientation.
Symmetry has numerous applications in our daily life.
For example, in art, architecture, textile technology, design relations, Rangoli, etc.