Question 4
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?
- When a folded paper is punched and then unfolded, the fold line acts as a line of symmetry (mirror line).
- The punched holes appear as symmetric pairs across each fold line.
- A single fold creates layers (resulting in holes from one punch), while two folds create layers (resulting in holes from one punch).
(a) Identify the line along which the paper was folded in figure (a).
Step 1 · Identify the Line of Symmetry
There are holes symmetric across the diagonal running from the top-left corner to the bottom-right corner.
Therefore, the paper was folded along the diagonal from top-left to bottom-right.
(a) Folded diagonally from top-left to bottom-right.
(b) Identify the line along which the paper was folded in figure (b).
Step 1 · Identify the Line of Symmetry
There are holes placed symmetrically across the horizontal center line.
Therefore, the paper was folded horizontally across the middle.
(b) Folded horizontally across the middle.
(c) Identify the line along which the paper was folded in figure (c).
Step 1 · Identify the Line of Symmetry
There are holes placed symmetrically across the vertical center line.
Therefore, the paper was folded vertically down the middle.
(c) Folded vertically down the middle.
(d) Figure (d) was created by punching a single hole. How was the paper folded?
Step 1 · Analyze the Folding Sequence
Figure (d) contains holes, one in each corner.
To produce holes from a single punch:
- Fold the paper along one diagonal.
- Fold the resulting triangle along the second diagonal to form a smaller triangle.
- Punching a hole through the corner where the four vertices overlap creates a hole at all four corners upon unfolding.
(d) Folded along both diagonals.
- Single vs. Double Fold: Assuming figure (d) was created with a single fold. A single fold yields at most holes per punch; holes require two folds ( layers).
- Horizontal vs. Vertical Lines: Confusing the orientation of horizontal (side-to-side) and vertical (up-and-down) axes of symmetry.
More questions in FIO
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, is the smallest angle of symmetry. What are the other angles of symmetry of this figure?
In a figure, is an angle of symmetry. The figure has two angles of symmetry less than . What is its smallest angle of symmetry?
Can we have a figure with rotational symmetry whose smallest angle of symmetry is:
a. ?
b. ?