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
Understand the Question
  • A circle is divided into 1212 equal sectors, each measuring 360∘12=30∘\dfrac{360^\circ}{12} = 30^\circ.
  • For a figure to have nn angles of rotational symmetry, the coloring pattern must repeat every 360∘n\dfrac{360^\circ}{n}, which corresponds to 12n\dfrac{12}{n} sectors.
  • Therefore, nn must be a factor of 1212 so that each color block divides the 1212 sectors evenly.

(i) Colour the sectors of the circle below so that the figure has 3 angles of symmetry

Step 1 · Determine Color Pattern for 3 Angles of Symmetry

For 33 angles of symmetry, the figure must look identical after rotating by 360∘3=120∘\dfrac{360^\circ}{3} = 120^\circ.

Number of sectors per repeating block: 120∘30∘=4 sectors\dfrac{120^\circ}{30^\circ} = 4\text{ sectors}Diagram 1

Color the sectors in blocks of 44 sequentially:

  • Sectors 1, 5, 9: Red
  • Sectors 2, 6, 10: Blue
  • Sectors 3, 7, 11: Green
  • Sectors 4, 8, 12: Yellow
Answer

(i) Repeat the color pattern every 44 sectors (rotational symmetry of 120∘120^\circ).

(ii) Colour the sectors of the circle below so that the figure has 4 angles of symmetry

Step 1 · Determine Color Pattern for 4 Angles of Symmetry

For 44 angles of symmetry, the figure must look identical after rotating by 360∘4=90∘\dfrac{360^\circ}{4} = 90^\circ.

Number of sectors per repeating block: 90∘30∘=3 sectors\dfrac{90^\circ}{30^\circ} = 3\text{ sectors}Diagram 2

Color the sectors in blocks of 33 sequentially:

  • Sectors 1, 4, 7, 10: Red
  • Sectors 2, 5, 8, 11: Blue
  • Sectors 3, 6, 9, 12: Green
Answer

(ii) Repeat the color pattern every 33 sectors (rotational symmetry of 90∘90^\circ).

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

Step 1 · Find Factors of 12 Sectors

The number of angles of symmetry must be a divisor of the total 1212 sectors (1,2,3,4,6,121, 2, 3, 4, 6, 12).

By creating non-trivial repeating patterns:

  • 2 angles of symmetry: Pattern repeats every 66 sectors (180∘180^\circ rotation)
  • 3 angles of symmetry: Pattern repeats every 44 sectors (120∘120^\circ rotation)
  • 4 angles of symmetry: Pattern repeats every 33 sectors (90∘90^\circ rotation)
  • 6 angles of symmetry: Pattern repeats every 22 sectors (60∘60^\circ rotation)
Answer

(iii) 2,3,4,2, 3, 4, or 66 angles of symmetry

Common Mistakes
  • Divisor Constraint: Choosing an order of symmetry that does not divide 1212 (e.g., 55 angles of symmetry). Since 1212 is not divisible by 55, the pattern cannot repeat uniformly.
  • Interval Confusion: For nn angles of symmetry, the pattern repeats every 12n\dfrac{12}{n} sectors, not every nn sectors.

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∘60^\circ is the smallest angle of symmetry. What are the other angles of symmetry of this figure?

Q17

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

Q18

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

a. 45∘45^\circ?

b. 17∘17^\circ?

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