Symmetry

Does a square have only one line of symmetry? Take a square piece of paper. By folding, find all its lines of symmetry.

Ink Blot Devils

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.

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.

Trace and cut out a copy of this figure. By rotating the cutout over this figure determine its angles of rotation.

Game

Draw a 6 by 6 grid. Two players take turns covering two adjacent squares by drawing a line. The line can be placed either way: horizontally or vertically. The lines cannot overlap. The game goes on till a player is not able to place any more lines. The player who is not able to place a line loses.

With what strategy can one play to win this game?

Do you see any line of symmetry in the figures at the start of the chapter? What about in the picture of the cloud?

For each of the following figures, identify the line(s) of symmetry if it exists.

Punching Game

The fold is a line of symmetry. Punch holes at different locations of a folded square sheet of paper using a punching machine and create different symmetric patterns.

In each of the following figures, a hole was punched in a folded square sheet of paper and then the paper was unfolded. Identify the line along which the paper was folded. Figure (d) was created by punching a single hole. How was the paper folded?

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

Here are some questions on paper cutting. Consider a vertical fold. We represent it this way:

Vertical Fold

Similarly, a horizontal fold is represented as follows:

Horizontal Fold

After each of the following cuts, predict the shape of the hole when the paper is opened. After you have made your prediction, make the cutouts and verify your answer.

Trace each figure and draw the lines of symmetry, if any:

Copy the following on a dot grid. For each figure draw two more lines to make a shape that has a line of symmetry.

Find the angles of symmetry for the given figures about the point marked •.

(a) (b) (c)

Which of the following figures have more than one angle of symmetry?

Give the order of rotational symmetry for each figure:

Colour the sectors of the circle below so that the figure has:

(i) 3 angles of symmetry

(ii) 4 angles of symmetry

(iii) What are the possible numbers of angles of symmetry you can obtain by colouring the sectors in different ways?

Draw two figures other than a circle and a square that have both reflection symmetry and rotational symmetry.

Draw, wherever possible, a rough sketch of:

a. A triangle with at least two lines of symmetry and at least two angles of symmetry.

b. A triangle with only one line of symmetry but not having rotational symmetry.

c. A quadrilateral with rotational symmetry but no reflection symmetry.

d. A quadrilateral with reflection symmetry but not having rotational symmetry.

In a figure, 60° is the smallest angle of symmetry. What are the other angles of symmetry of this figure?

In a figure, 60° is an angle of symmetry. The figure has two angles of symmetry less than 60°. What is its smallest angle of symmetry?

Can we have a figure with rotational symmetry whose smallest angle of symmetry is:

a. 45°?

b. 17°?

What about the butterfly? No doubt, the colours are very attractive. But what else about the butterfly appeals to you?

Can you see what repeats in the beautiful rangoli figure?

What about the pinwheel? Can you spot which pattern is repeating?

Hint: Look at the hexagon first.

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 recognise 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?

What are the symmetries that you see in these beautiful structures?

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?

Context: Consider a square with its corners labeled A, B, C and D, and its vertical line of symmetry as shown in the figure.

Q. 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?

In these two figures, a sheet of paper is folded and a cut is made along the dotted line shown. Draw a sketch of how the paper will look when unfolded.

Do you see a line of symmetry in this figure? What is it?

5 Suppose you have to get each of these shapes with some folds and a single straight cut. How will you do it?

a. The hole in the centre is a square.

b. The hole in the centre is a square.

Note: For the above two questions, check if the 4-sided figures in the centre satisfy both the properties of a square.

6. How many lines of symmetry do these shapes have?

Find the lines of symmetry for the kolam below.

Draw the following.

a. A triangle with exactly one line of symmetry. b. A triangle with exactly three lines of symmetry. c. A triangle with no line of symmetry.

Is it possible to draw a triangle with exactly two lines of symmetry?

Draw the following. In each case, the figure should contain at least one curved boundary.

a. A figure with exactly one line of symmetry. b. A figure with exactly two lines of symmetry. c. A figure with exactly four lines of symmetry.

Copy the following on squared paper. Complete them so that the blue line is a line of symmetry. Problem (a) has been done for you.

Hint: For (c) and (f), see if rotating the book helps!

Copy the following drawing on squared paper. Complete each one of them so that the resulting figure has the two blue lines as lines of symmetry.

Do you know of any other shape that has exactly four angles of symmetry?

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.

How will you modify the figure above so that it has only two 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?

However, can anything in the figure be changed to make it have 3 angles of symmetry?

Can you draw a figure with radial arms that has a) exactly 5 angles of symmetry, b) 6 angles of symmetry? Also find the angles of symmetry in each case.

Hint: Use 5 radial arms for the first case. What should the angle between two adjacent radial arms be?

Consider a figure with radial arms having exactly 7 angles of symmetry. What will be its smallest angle of symmetry? Is the number of degrees a whole number in this case? If not, express it as a mixed fraction.

In each case, the angles are the multiples of the smallest angle. You may wonder and ask if this will always happen. What do you think?

True or False

• Every figure will have 360 degrees as an angle of symmetry.
• If the smallest angle of symmetry of a figure is a natural number in degrees, then it is a factor of 360.

Like wheels, we can find other objects around us having rotational symmetry. Find them.