Question 11
5 Suppose you have to get each of these shapes with some folds and a single straight cut. How will you do it?
a. The hole in the centre is a square.
b. The hole in the centre is a square.
Note: For the above two questions, check if the 4-sided figures in the centre satisfy both the properties of a square.
- Using lines of symmetry, folding a square paper multiple times allows us to create symmetric geometric shapes in the center with a single straight cut.
- The central point where all fold lines intersect corresponds to the exact center of the unfolded paper.
- The orientation of the folds (horizontal/vertical lines of symmetry vs. diagonal lines of symmetry) determines the alignment of the resulting central square hole.
a. The hole in the centre is a square.
Step 1 · Fold along horizontal and vertical lines of symmetry
- Take a square sheet of paper.
- Fold the paper in half along its length, then in half again along its breadth to form a smaller square.
- The folded corner where all folds intersect is the center of the sheet.
- Make a straight cut at this corner with cut edges parallel to the edges of the sheet.
- Unfolding the sheet gives a square hole in the center.
a. Fold the square paper in half horizontally and vertically, then make a straight cut at the corner where all folds meet parallel to the outer edges.
b. The hole in the centre is a square.
Step 1 · Fold along diagonal lines of symmetry
- Take a square sheet of paper.
- Fold the paper in half along one diagonal to form a triangle.
- Fold it in half again along the second diagonal to get a smaller triangle.
- The corner where all folds meet corresponds to the center of the sheet.
- Make a single straight cut across this closed corner.
- Unfolding the sheet reveals a square (diamond-oriented) hole at the center.
b. Fold the square paper in half along both diagonals, then make a single straight cut across the central corner where all folds meet.
- Cutting the Wrong Corner: Cutting the open outer edges instead of the closed central corner where all the folds intersect will remove the corners of the paper rather than creating a hole in the center.
- Unequal Angle Cut: For part (b), the cut must be symmetric across the folded edge so that all four sides and angles of the resulting shape are equal (forming a true square).
More questions in IT
What about the butterfly? No doubt, the colours are very attractive. But what else about the butterfly appeals to you?
Can you see what repeats in the beautiful rangoli figure?
What about the pinwheel? Can you spot which pattern is repeating?
Hint: Look at the hexagon first.
Now, can you say what figure repeats along each side of the hexagon? What is the shape of the figure that is stuck to each side? Do you recognise it? How do these shapes move as you move along the boundary of the hexagon? What about the other pictures—what is it about those structures that appeals to you and what are the patterns in those structures that repeat?
What are the symmetries that you see in these beautiful structures?
Is there any other way to fold the square so that the two halves overlap? How many lines of symmetry does the square shape have?
Thus, figures can have multiple lines of symmetry. The figures below also have multiple lines of symmetry. Can you find them all?
We saw that the diagonal of a square is also a line of symmetry. Let us take a rectangle that is not a square. Is its diagonal a line of symmetry?
First, see the rectangle and answer this question. Then, take a rectangular piece of paper and check if the two parts overlap by folding it along its diagonal. What do you observe?
Context: Consider a square with its corners labeled , , and , and its vertical line of symmetry as shown in the figure.
Q. What if we reflect along the diagonal from to ? Where do points , , and go? What if we reflect along the horizontal line of symmetry?
In these two figures, a sheet of paper is folded and a cut is made along the dotted line shown. Draw a sketch of how the paper will look when unfolded.
Do you see a line of symmetry in this figure? What is it?
5 Suppose you have to get each of these shapes with some folds and a single straight cut. How will you do it?
a. The hole in the centre is a square.
b. The hole in the centre is a square.
Note: For the above two questions, check if the 4-sided figures in the centre satisfy both the properties of a square.
- How many lines of symmetry do these shapes have?
Find the lines of symmetry for the kolam below.
Draw the following.
a. A triangle with exactly one line of symmetry. b. A triangle with exactly three lines of symmetry. c. A triangle with no line of symmetry.
Is it possible to draw a triangle with exactly two lines of symmetry?
Draw the following. In each case, the figure should contain at least one curved boundary.
a. A figure with exactly one line of symmetry. b. A figure with exactly two lines of symmetry. c. A figure with exactly four lines of symmetry.
Copy the following on squared paper. Complete them so that the blue line is a line of symmetry. Problem (a) has been done for you.
Hint: For (c) and (f), see if rotating the book helps!
Copy the following drawing on squared paper. Complete each one of them so that the resulting figure has the two blue lines as lines of symmetry.
Do you know of any other shape that has exactly four angles of symmetry?
Consider this figure, a picture with 4 radial arms. How many angles of symmetry does it have? What are they? Note that the angle between adjacent central dotted lines is .
Can you change the angles between the radial arms so that the figure still has 4 angles of symmetry? Try drawing it.
How will you modify the figure above so that it has only two angles of symmetry?
Can we get a figure having exactly 3 angles of symmetry? Can you use radial arms for this?
Let us try with 3 radial arms as in the figure below. How many angles of symmetry does it have and what are they?
However, can anything in the figure be changed to make it have 3 angles of symmetry?
Can you draw a figure with radial arms that has a) exactly 5 angles of symmetry, b) 6 angles of symmetry? Also find the angles of symmetry in each case.
Hint: Use 5 radial arms for the first case. What should the angle between two adjacent radial arms be?
Consider a figure with radial arms having exactly angles of symmetry. What will be its smallest angle of symmetry? Is the number of degrees a whole number in this case? If not, express it as a mixed fraction.
In each case, the angles are the multiples of the smallest angle. You may wonder and ask if this will always happen. What do you think?
True or False
- Every figure will have as an angle of symmetry.
- If the smallest angle of symmetry of a figure is a natural number in degrees, then it is a factor of .
Like wheels, we can find other objects around us having rotational symmetry. Find them.