Symmetry | FIO

Question 10

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

(a) (b) (c)

Question diagram 1
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Solution
Understand the Question
  • The angle of symmetry (or angle of rotational symmetry) is the smallest angle by which a figure can be rotated about a fixed point so that it coincides with its original position.
  • A full turn is 360∘360^\circ.
  • If a figure fits onto itself nn times during a complete 360∘360^\circ rotation, its angle of symmetry is given by: Angle of symmetry=360∘n\text{Angle of symmetry} = \dfrac{360^\circ}{n}

(a) Find the angle of symmetry for figure (a) about the point marked •.

Step 1 · Find the Angle of Symmetry for Figure (a)

Diagram 1

Figure (a) is a symmetric cross shape about its center point •.

Rotating by 90∘90^\circ makes the figure look identical to the original orientation. It coincides with itself 44 times in a complete 360∘360^\circ turn.

Angle of symmetry=360∘Number of times it looks the same=360∘4=90∘\begin{aligned} \text{Angle of symmetry} &= \dfrac{360^\circ}{\text{Number of times it looks the same}} \\[0.6em] &= \dfrac{360^\circ}{4} \\[0.6em] &= 90^\circ \end{aligned}
Answer

(a) 90∘90^\circ

(b) Find the angle of symmetry for figure (b) about the point marked •.

Step 1 · Find the Angle of Symmetry for Figure (b)

Diagram 2

Figure (b) consists of a vertical line with two squares at its ends pointing in the same direction.

  • A rotation of 90∘90^\circ or 180∘180^\circ does not restore the original orientation.
  • The figure only matches itself after a complete full turn of 360∘360^\circ (11 time).
Angle of symmetry=360∘Number of times it looks the same=360∘1=360∘\begin{aligned} \text{Angle of symmetry} &= \dfrac{360^\circ}{\text{Number of times it looks the same}} \\[0.6em] &= \dfrac{360^\circ}{1} \\[0.6em] &= 360^\circ \end{aligned}
Answer

(b) 360∘360^\circ

(c) Find the angle of symmetry for figure (c) about the point marked •.

Step 1 · Find the Angle of Symmetry for Figure (c)

Diagram 3

Figure (c) is shaped like the letter 'H' with the center point • on the horizontal segment.

  • A rotation of 90∘90^\circ turns it into an 'I' shape (horizontal).
  • A rotation of 180∘180^\circ restores the original 'H' shape.
  • The figure coincides with itself 22 times in a complete 360∘360^\circ turn.
Angle of symmetry=360∘Number of times it looks the same=360∘2=180∘\begin{aligned} \text{Angle of symmetry} &= \dfrac{360^\circ}{\text{Number of times it looks the same}} \\[0.6em] &= \dfrac{360^\circ}{2} \\[0.6em] &= 180^\circ \end{aligned}
Answer

(c) 180∘180^\circ

Common Mistakes
  • Assuming 180∘180^\circ for Figure (b): In figure (b), both squares point to the same side. Rotating by 180∘180^\circ moves the top square to the bottom and inverts its direction relative to the original orientation, meaning it only has an angle of symmetry of 360∘360^\circ.
  • Confusing Line and Rotational Symmetry: Line symmetry involves folding across an axis, whereas rotational symmetry requires rotating the shape about a single center point.

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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