Class 7 — Mathematics (NCERT)

Symmetry

Class 7

✓By the end of this lesson students will be able to define and identify lines of symmetry in various 2D shapes.
✓By the end of this lesson students will be able to define and identify rotational symmetry in various 2D shapes.
✓By the end of this lesson students will be able to determine the order and angle of rotational symmetry for given figures.
✓By the end of this lesson students will be able to distinguish between line symmetry and rotational symmetry.
✓By the end of this lesson students will be able to draw figures with a specified number of lines of symmetry or a given order of rotational symmetry.

Key concepts

Line of Symmetry

A line of symmetry (also called an axis of symmetry) is a line that divides a figure into two identical halves such that if the figure is folded along this line, the two halves coincide exactly. The two halves are mirror images of each other. A figure can have one, more than one, or no lines of symmetry.

Rotational Symmetry

A figure is said to have rotational symmetry if it looks exactly the same after being rotated by some angle less than 360 degrees about a fixed point. This fixed point is called the centre of rotation. The angle by which the object rotates to achieve rotational symmetry is called the angle of rotation. The number of times a figure looks exactly the same during a full rotation of 360 degrees is called the order of rotational symmetry.

Angle of Rotational Symmetry = 360 degrees / Order of Rotational Symmetry

Key facts to remember

1A line of symmetry divides a figure into two identical mirror halves.
2A figure can have 0, 1, 2, 3, 4, or even infinite lines of symmetry.
3A circle has infinite lines of symmetry.
4Rotational symmetry occurs when a figure looks the same after rotation by an angle less than 360 degrees about a fixed point (centre of rotation).
5The order of rotational symmetry is the number of times a figure looks identical during a 360-degree rotation.
6The angle of rotational symmetry is calculated as 360 degrees divided by the order of rotational symmetry.
7Every object has rotational symmetry of order 1 (at 360 degrees), but for 'true' rotational symmetry, we usually consider an order greater than 1.

Worked examples

Example 1

Draw all lines of symmetry for a rectangle and state the number of lines of symmetry.

IStep 1: Draw a rectangle, say ABCD.

IIStep 2: Identify the line passing through the midpoints of the longer sides (AD and BC). Let's call this line L1. If we fold the rectangle along L1, the two halves will coincide exactly. So, L1 is a line of symmetry.

IIIStep 3: Identify the line passing through the midpoints of the shorter sides (AB and CD). Let's call this line L2. If we fold the rectangle along L2, the two halves will coincide exactly. So, L2 is another line of symmetry.

IVStep 4: Check for diagonal lines. If we fold a rectangle along its diagonal, the two halves do not coincide exactly. Hence, diagonals are not lines of symmetry for a rectangle.

VStep 5: Count the identified lines of symmetry.

Answer

A rectangle has 2 lines of symmetry.

Always use a ruler and pencil to draw lines of symmetry accurately.

Example 2

Determine the order and angle of rotational symmetry for a square.

IStep 1: Draw a square. Mark its centre (the point where its diagonals intersect).

IIStep 2: Imagine rotating the square about its centre. After a rotation of 90 degrees, the square will look exactly the same as its original position.

IIIStep 3: Rotate it by another 90 degrees (total 180 degrees). It will again look exactly the same.

IVStep 4: Rotate it by another 90 degrees (total 270 degrees). It will again look exactly the same.

VStep 5: Rotate it by another 90 degrees (total 360 degrees). It returns to its original position.

VIStep 6: Count the number of times the square looks exactly the same during a full 360-degree rotation (excluding the 360-degree position as a distinct new position, but including it in the count of identical appearances). This count is 4.

VIIStep 7: The order of rotational symmetry is 4.

VIIIStep 8: Calculate the angle of rotational symmetry using the formula: Angle = 360 degrees / Order.

Answer

A square has an order of rotational symmetry of 4 and an angle of rotational symmetry of 90 degrees (360/4 = 90).

You can trace the shape on a separate sheet and rotate it around a pin at the centre to visualise the rotations.

Example 3

For an equilateral triangle, find the number of lines of symmetry and its order of rotational symmetry.

IStep 1: Draw an equilateral triangle.

IIStep 2: To find lines of symmetry: An equilateral triangle has three equal sides and three equal angles. A line drawn from each vertex to the midpoint of the opposite side will divide the triangle into two identical halves. There are 3 such lines.

IIIStep 3: To find rotational symmetry: Identify the centre of the equilateral triangle (the point where the medians intersect).

IVStep 4: Rotate the triangle about its centre. Since all angles are 60 degrees, and the triangle has 3 identical 'faces' from the centre, it will look the same after a rotation of 360/3 = 120 degrees.

VStep 5: After 120 degrees, it looks the same. After another 120 degrees (total 240 degrees), it looks the same. After another 120 degrees (total 360 degrees), it returns to its original position.

VIStep 6: The number of times it looks the same during a 360-degree rotation is 3. This is the order of rotational symmetry.

VIIStep 7: The angle of rotational symmetry is 360 degrees / 3 = 120 degrees.

Answer

An equilateral triangle has 3 lines of symmetry and an order of rotational symmetry of 3 (with an angle of 120 degrees).

Regular polygons always have the same number of lines of symmetry as their number of sides, and their order of rotational symmetry is also equal to their number of sides.

Common mistakes

✗Confusing lines of symmetry with diagonals, especially in shapes like rectangles where diagonals are not lines of symmetry.
✗Incorrectly identifying the centre of rotation for a figure.
✗Forgetting to count the original position when determining the order of rotational symmetry, leading to an incorrect order.
✗Assuming that all figures with line symmetry also have rotational symmetry, or vice-versa (e.g., an isosceles triangle has line symmetry but usually not rotational symmetry of order > 1).
✗Not understanding that a figure must look *exactly* the same, not just similar, after rotation or folding.

Exam tips

★Always use a ruler and a sharp pencil to draw shapes and lines of symmetry accurately in your answer sheet.
★For rotational symmetry questions, if allowed, use tracing paper and a pin to physically rotate the shape and visualise its positions.
★When asked for rotational symmetry, always state both the 'order' and the 'angle' of rotational symmetry.
★Practice identifying symmetry in various real-world objects, letters of the alphabet, and geometric shapes to improve your understanding.

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