Symmetry | FIO

Question 7

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.

Question diagram 1
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Solution
Understand the Question
  • When paper is folded and cut, every fold line acts as an axis of symmetry (mirror line).
  • Unfolding the paper mirrors the cut across each fold line, repeating the cut shape symmetrically.
  • A cut made at the corner where all folds meet forms a central hole in the unfolded sheet, while cuts along the outer edges form notches along the perimeter.

(a) Predict the shape of the hole when the paper is opened after cut (a).

Step 1 · Predict cutout shape for (a)

A square paper is folded in half (creating 22 layers), and a wavy shape is cut along the folded crease.Diagram 1

When unfolded, the cut mirrors across the single vertical line of symmetry to form a symmetrical decorative or butterfly-like shape.

Answer

(a) A symmetric butterfly-like or decorative shape

(b) Predict the shape of the hole when the paper is opened after cut (b).

Step 1 · Predict cutout shape for (b)

The paper is folded into quarters, then diagonally and inwards to create 88 layers. A straight cut is made across the tip where all folds meet.Diagram 2

When unfolded, the cut is repeated 88 times symmetrically around the center, forming an 88-sided regular hole (an octagon).

Answer

(b) A symmetric octagonal shape

(c) Predict the shape of the hole when the paper is opened after cut (c).

Step 1 · Predict cutout shape for (c)

The square sheet is folded into quarters and then folded vertically again (88 layers). Two square cuts are made:

  1. A cut at the corner where all folds meet (the center of the paper).
  2. A cut along the open edge.Diagram 3

When unfolded, the central corner cut forms a square hole at the center, and the edge cut forms a square notch at the outer edge.

Answer

(c) A square hole at the center and a square cutout at the bottom

(d) Predict the shape of the hole when the paper is opened after cut (d).

Step 1 · Predict cutout shape for (d)

A square paper is folded in half, and a T\text{T}-shaped cutout is made along the folded edge.Diagram 4

When unfolded, the T\text{T}-shape mirrors across the crease, producing two symmetrical rectangular cutouts on opposite sides.

Answer

(d) Two rectangular cutouts on opposite sides

Common Mistakes
  • Confusing Center Corners with Outer Edges: Cutting the corner where all folds meet creates a hole in the center of the unfolded paper, whereas cutting an outer edge creates a notch on the boundary.
  • Miscounting Symmetry Repetitions: Each fold doubles the number of layers (1 fold →2\rightarrow 2 layers, 2 folds →4\rightarrow 4 layers, 3 folds →8\rightarrow 8 layers). The cut is replicated once per layer.

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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