Table of contents | |
Introduction | |
Reflection and Symmetry | |
Section | |
Planes of Symmetry | |
Rotational Symmetry | |
No Rotational Symmetry | |
Nets |
(ii) Also the length of = length of
Also length of = length of
Also length of = length of
and ∠BAC = ∠B'A'C', ∠ABC = ∠A'B'C', ∠BCA = ∠B'C′A'.
(iii) AA', BB' and CC' are perpendicular to the mirror line and are bisected by the mirror line.
Let ABCDE be a figure given to us whose mirror image is to be drawn.
We draw perpendiculars from the vertices of the given figure to the mirror line. AP, E4, DR, CS and BT are drawn perpendicular to the mirror line.
Let ABCDE be the given part of the figure and p, and q be the two lines of symmetry.
.
When we make a straight cut through a solid, we say that the solid is cut by a plane, i.e., the section is flat.
Imagine that a cuboid is cut into two pieces by a plane as shown below.
Now take one of the pieces and put the cut face against a mirror.
Example 1: Draw a line segment AB having a fixed length. Mark its mid-point O. Fix a pin through O. Now, keeping the middle point fixed, rotate the line segment AB about O. After a rotation of 180, you will find that the line segment takes position B1A1. Thus, the new position B1A1 fits into the original line segment AB.
Another rotation of 180 about the central point O gives A1B1, a new position A2B2, once again fitting into the original line segment AB, A2 falling on A and B2 falling on B.
The midpoint of the line segment is the centre of rotation.
The point around which the figure is rotated is called the centre of rotation.
Thus, A figure has rotational symmetry, if there is a central point around which the figure is rotated through a certain number of degrees (less than 180) and the figure still looks the same. The central point is called the centre of rotation. The smallest angle we need to turn the figure to get a similar figure again is called the angle of rotation.
The number of times a shape will fit onto itself in one complete rotation is called the order of the rotational symmetry.
Edurev Tips: A full rotation does not mean that a figure has rotational symmetry because every shape could fit exactly into itself after a full (360º) rotation.
Example 2: Rotate an equilateral triangle about its centroid (the point where the medians meet) through an angle of 120º. After a rotation of 120º, 240º, 360º, the triangle gets three new positions of A1B1C1, A2B2C2 and A3B3C3 falling on BCA, CAB and ABC respectively.
Thus, the triangle matches itself three times during a rotation of 360º. Therefore, it has rotational symmetry of order 3.
Here, the order of rotational symmetry = 360º/120º = 3
Example 3: Rotate a square about its centre (the point where its diagonals meet), through an angle of 90. After each rotation of 90, the new square fits into the original square, i.e., the square matches itself after a rotation of 90, 180, 270 and 360, i.e., 4 times during one complete rotation of 360.
∴ Angle of rotation = 90º
Here, the order of rotation = 360º/90º = 4
In general, we have
Order of rotational symmetry = 360º/the smallest angle of rotation
Example 4: A rectangle has rotational symmetry about its centre (the point where its diagonals meet)
Angle of rotation = 180º
∴ Order of rotational symmetry = 360º/180º = 2
Example 5: A regular pentagon (a polygon having 5 equal sides), a design of flower design with 5 petals are the figures having rotational symmetry of order 5.
Example 6: A regular hexagon (a polygon having 6 equal sides) has rotational symmetry of order 6 and angle of rotation = 60º.
A regular octagon (a polygon having 8 equal sides) has rotational symmetry of order 8 and angle of rotation = 45º.
Some figures have both types of symmetry, linear and rotational.
Consider the following examples :
Only Linear Symmetry
There are some figures which have only one line symmetry but have no rotational symmetry.
For example (a) Letters of the English alphabet A, C, D, E, M, T, U, 9, W, Y
(b) Isosceles triangle.
Only Rotational Symmetry
We have some figures that have only rotational symmetry but no linear symmetry. For example, letters of the English alphabet N, S, and Z.
If we join together six identical squares, edge to edge, we get a cube or the outside surface of the cube.
We can avoid a lot of unnecessary sticking if we join some squares together before cutting them out.
Suppose we want to make a cube out of cardboard or a piece of paper. We need a pattern giving us the shape of the cardboard or the piece of paper to make the cube.
Figure (a) below shows, the shape of the pattern of six squares. When this shape is folded along the edges, a cube is formed as shown.
For example, a cube can also be made from the nets shown below.
However, not all arrangements of six squares will work as shown below :
It is easy to make a solid shape from a piece of paper or cardboard by first drawing a net of the faces of the solid. The net can be made up of different plane shapes (rectangles, squares, triangles).
The figure below shows the net of a cuboid of dimensions 5 cm × 3 cm × 2 cm.
The net for a cylinder without a top and a bottom is shown. The length of the net is equal to the circumference of the cylinder.
The net of a cone has one circular base and one curved surface.
58 videos|112 docs|40 tests
|
1. What is symmetry and why is it important in geometry? |
2. What are planes of symmetry? |
3. How can we identify rotational symmetry in a shape? |
4. What are some examples of shapes with no rotational symmetry? |
5. What are nets in geometry and how are they related to symmetry? |
|
Explore Courses for Class 5 exam
|