Question 14
Draw two figures other than a circle and a square that have both reflection symmetry and rotational symmetry.
- Reflection Symmetry (Line Symmetry): A figure has reflection symmetry if it can be folded along a line such that both halves match exactly.
- Rotational Symmetry: A figure has rotational symmetry if it looks identical to its original position after being rotated about its centre by an angle of less than .
- Regular polygons (such as an equilateral triangle or a regular hexagon) as well as figures like a rectangle or rhombus have both types of symmetry.
Step 1 · Equilateral Triangle
- Reflection Symmetry: It has lines of symmetry. Each line connects a vertex to the midpoint of the opposite side.
- Rotational Symmetry: Rotating the triangle about its centre by or makes it look identical to the original. It has rotational symmetry of order .
Step 2 · Regular Hexagon
- Reflection Symmetry: It has lines of symmetry ( lines connecting opposite vertices and lines connecting midpoints of opposite sides).
- Rotational Symmetry: Rotating it about its centre by multiples of makes it look identical to the original, giving it rotational symmetry of order .
1. Equilateral triangle 2. Regular hexagon
- Trivial Rotation (): Every shape looks identical after a full turn. For a figure to have rotational symmetry, the angle of rotation must be strictly less than (order ).
- Regular vs Irregular Polygons: Only regular triangles (equilateral) and regular hexagons have all lines of symmetry and rotational symmetry; scalene or irregular versions do not.
- Other Valid Shapes: Shapes such as a rectangle (2 lines of symmetry, rotational order 2) and a rhombus (2 lines of symmetry, rotational order 2) are also correct valid examples.
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. ?