Symmetry | FIO

Question 13

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?

Question diagram 1
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Solution

FIO-13

Chapter: SYMMETRY
Class: 6 (Class 6)
Category: figure_it_out

Question

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?

Question diagram(s):

Question diagram

The number of angles of symmetry must be a factor of the total number of sectors.

Step 1 — Coloring for 3 angles of symmetry

The circle has 12 equal sectors. Each sector covers 30 degrees. We want 3 angles of symmetry. This means the figure must look the same after rotating by 120 degrees. Let us find how many sectors this rotation covers.

120∘/30∘120^\circ / 30^\circ

4 sectors\boxed{4 \text{ sectors}}

So, the pattern of colors must repeat every 4 sectors. Let us number the sectors from 1 to 12 clockwise. We can color sector 1 red. Then sector 5 must be red. Sector 9 must be red. We can color sector 2 blue. Then sector 6 must be blue. Sector 10 must be blue. We can color sector 3 green. Then sector 7 must be green. Sector 11 must be green. We can color sector 4 yellow. Then sector 8 must be yellow. Sector 12 must be yellow. This coloring makes the figure look the same after every rotation of 120 degrees.

Diagram 1

Step 2 — Coloring for 4 angles of symmetry

We want 4 angles of symmetry. This means the figure must look the same after rotating by 90 degrees. Let us find how many sectors this rotation covers.

90∘/30∘90^\circ / 30^\circ

3 sectors\boxed{3 \text{ sectors}}

So, the pattern of colors must repeat every 3 sectors. Let us number the sectors from 1 to 12 clockwise. We can color sector 1 red. Then sector 4 must be red. Sector 7 must be red. Sector 10 must be red. We can color sector 2 blue. Then sector 5 must be blue. Sector 8 must be blue. Sector 11 must be blue. We can color sector 3 green. Then sector 6 must be green. Sector 9 must be green. Sector 12 must be green. This coloring makes the figure look the same after every rotation of 90 degrees.

Diagram 2

Step 3 — Possible numbers of angles of symmetry

The number of angles of symmetry must be a divisor of the total number of sectors. The total number of sectors is 12. The divisors of 12 are 1, 2, 3, 4, 6, 12. We can achieve different numbers of angles of symmetry. To get 2 angles of symmetry, the pattern of colors repeats every 6 sectors. This means it looks the same after a 180 degree rotation. To get 3 angles of symmetry, the pattern of colors repeats every 4 sectors. This means it looks the same after a 120 degree rotation. To get 4 angles of symmetry, the pattern of colors repeats every 3 sectors. This means it looks the same after a 90 degree rotation. To get 6 angles of symmetry, the pattern of colors repeats every 2 sectors. This means it looks the same after a 60 degree rotation. The possible numbers of angles of symmetry are 2, 3, 4, or 6.

Answer

(i) Will look same after every rotation of 120°. (ii) Will look same after every rotation of 90°. (iii) Four ways are possible (2, 3, 4, or 6 angles of symmetry).

More questions in FIO

Q1

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

Q2

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

Q3

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.

Q4

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?

Q5

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

Q6

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

Q7

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.

Q8

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

Q9

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

Q10

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

(a) (b) (c)

Q11

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

Q12

Give the order of rotational symmetry for each figure:

Q13

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?

Q14

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

Q15

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.

Q16

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

Q17

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?

Q18

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

a. 45°?

b. 17°?

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