Class 7 Mathematics Notes Chapter 14 (Symmetry) – Mathematics Book
Detailed Notes with MCQs of Chapter 14, 'Symmetry'. This is an important topic, not just for your Class 7 understanding, but also because concepts of symmetry appear frequently in various competitive and government exams, often testing spatial reasoning and geometric properties. Pay close attention to the definitions and examples.
Chapter 14: Symmetry - Detailed Notes for Exam Preparation
1. Introduction to Symmetry
- Symmetry: In simple terms, symmetry refers to a sense of balanced and proportionate similarity that is found in two halves of an object, meaning one half is the mirror image of the other half. It can also refer to an object looking the same after undergoing a transformation like rotation.
- Symmetrical objects are aesthetically pleasing and structurally stable in many cases (e.g., architecture, nature).
- Two main types of symmetry are studied in this chapter:
- Line Symmetry (or Reflectional Symmetry)
- Rotational Symmetry
2. Line Symmetry (Reflectional Symmetry)
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Definition: A figure has line symmetry if a line can be drawn dividing the figure into two identical parts such that if the figure is folded along this line, the two parts coincide exactly.
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Line of Symmetry (or Axis of Symmetry): The imaginary line along which the figure is folded is called the line of symmetry.
-
Properties:
- A figure may have no line of symmetry, one line of symmetry, or multiple lines of symmetry.
- The line of symmetry acts like a mirror (plane mirror). One half of the figure is the reflection of the other half.
- Every point on one side of the line of symmetry has a corresponding image point on the other side, equidistant from the line. The segment joining a point and its image is perpendicular to the line of symmetry.
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Examples of Lines of Symmetry in Shapes:
- Line Segment: Has one line of symmetry – its perpendicular bisector.
- Angle (with equal arms): Has one line of symmetry – the angle bisector.
- Isosceles Triangle: Has one line of symmetry – the median/altitude from the vertex angle to the base.
- Equilateral Triangle: Has three lines of symmetry – the three angle bisectors/medians/altitudes.
- Rectangle: Has two lines of symmetry – the lines joining the midpoints of opposite sides. (Diagonals are NOT lines of symmetry).
- Square: Has four lines of symmetry – two lines joining midpoints of opposite sides and the two diagonals.
- Rhombus: Has two lines of symmetry – its diagonals.
- Parallelogram: Generally has no line of symmetry (unless it's a rectangle, square, or rhombus).
- Kite: Has one line of symmetry – the longer diagonal (or the diagonal connecting the vertices between equal sides).
- Isosceles Trapezium: Has one line of symmetry – the line joining the midpoints of the parallel sides.
- Circle: Has infinite lines of symmetry – every diameter is a line of symmetry.
- Regular Polygons: A regular polygon with 'n' sides has 'n' lines of symmetry.
- Regular Pentagon (5 sides): 5 lines of symmetry.
- Regular Hexagon (6 sides): 6 lines of symmetry.
- Letters of the Alphabet:
- Vertical Line of Symmetry: A, H, I, M, O, T, U, V, W, X, Y
- Horizontal Line of Symmetry: B, C, D, E, H, I, K, O, X
- Both Horizontal and Vertical: H, I, O, X
- No Line of Symmetry: F, G, J, L, N, P, Q, R, S, Z
3. Rotational Symmetry
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Definition: A figure has rotational symmetry if it looks exactly the same as its original position after being rotated through an angle less than 360° about a fixed point.
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Centre of Rotation: The fixed point about which the rotation occurs.
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Angle of Rotation: The smallest angle (greater than 0° and less than or equal to 360°) through which the figure must be rotated to look identical to its original position.
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Order of Rotational Symmetry: The number of times a figure fits onto itself (looks identical) during a full rotation (360°).
- Order 1 means the shape looks the same only after a full 360° rotation (effectively, no rotational symmetry apart from the trivial full turn).
- Calculation: Order = 360° / (Angle of Rotation)
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Examples of Rotational Symmetry:
- Square:
- Centre of Rotation: Intersection point of diagonals.
- Angle of Rotation: 90°
- Order of Rotational Symmetry: 4 (at 90°, 180°, 270°, 360°)
- Rectangle:
- Centre of Rotation: Intersection point of diagonals.
- Angle of Rotation: 180°
- Order of Rotational Symmetry: 2 (at 180°, 360°)
- Equilateral Triangle:
- Centre of Rotation: Centroid (intersection point of medians).
- Angle of Rotation: 120°
- Order of Rotational Symmetry: 3 (at 120°, 240°, 360°)
- Regular Hexagon:
- Centre of Rotation: Centre of the hexagon.
- Angle of Rotation: 60° (360°/6)
- Order of Rotational Symmetry: 6
- Circle:
- Centre of Rotation: Centre of the circle.
- Angle of Rotation: Any angle.
- Order of Rotational Symmetry: Infinite.
- Parallelogram:
- Centre of Rotation: Intersection point of diagonals.
- Angle of Rotation: 180°
- Order of Rotational Symmetry: 2
- Rhombus:
- Centre of Rotation: Intersection point of diagonals.
- Angle of Rotation: 180°
- Order of Rotational Symmetry: 2
- Letters of the Alphabet: (Consider standard fonts)
- Order 2: H, I, N, O, S, X, Z (Angle 180°)
- Order > 1 (excluding Order 2): O (Infinite, like a circle)
- Order 1 (No rotational symmetry): A, B, C, D, E, F, G, J, K, L, M, P, Q, R, T, U, V, W, Y
- Square:
4. Relationship Between Line and Rotational Symmetry
- Some shapes have only line symmetry (e.g., Isosceles Triangle).
- Some shapes have only rotational symmetry (e.g., Parallelogram - generally).
- Some shapes have both line and rotational symmetry (e.g., Square, Equilateral Triangle, Circle, Regular Polygons).
- Some shapes have neither line nor rotational symmetry (e.g., Scalene Triangle, general Quadrilateral).
Key Exam Points:
- Be able to identify lines of symmetry in given figures quickly.
- Know the number of lines of symmetry for standard geometric shapes, especially regular polygons.
- Be able to determine the centre, angle, and order of rotational symmetry.
- Distinguish between shapes having only line, only rotational, both, or neither type of symmetry.
- Apply these concepts to letters of the alphabet and everyday objects.
Multiple Choice Questions (MCQs)
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How many lines of symmetry does a rectangle have?
(a) 1
(b) 2
(c) 4
(d) Infinite -
What is the order of rotational symmetry for a square?
(a) 1
(b) 2
(c) 3
(d) 4 -
Which of the following letters has only a horizontal line of symmetry (in standard form)?
(a) A
(b) B
(c) H
(d) M -
A regular pentagon has an angle of rotational symmetry of:
(a) 90°
(b) 60°
(c) 72°
(d) 120° -
Which shape generally has rotational symmetry of order 2 but no line of symmetry?
(a) Square
(b) Rhombus
(c) Parallelogram
(d) Kite -
The number of lines of symmetry in a circle is:
(a) 0
(b) 1
(c) 4
(d) Infinite -
What is the order of rotational symmetry for the letter 'H'?
(a) 1
(b) 2
(c) 4
(d) Infinite -
A triangle with only one line of symmetry is called:
(a) Equilateral
(b) Scalene
(c) Isosceles
(d) Right-angled -
If a figure has an angle of rotational symmetry of 180°, what is its order of rotational symmetry?
(a) 1
(b) 2
(c) 3
(d) 4 -
Which of these figures has both line symmetry and rotational symmetry of order greater than 1?
(a) Isosceles Triangle
(b) Parallelogram
(c) Scalene Triangle
(d) Equilateral Triangle
Answer Key:
- (b) 2
- (d) 4
- (b) B (Also C, D, E, K have only horizontal)
- (c) 72° (360° / 5 = 72°)
- (c) Parallelogram
- (d) Infinite
- (b) 2
- (c) Isosceles
- (b) 2 (360° / 180° = 2)
- (d) Equilateral Triangle (3 lines of symmetry, rotational order 3)
Study these notes carefully. Visualizing the shapes and their transformations (folding and rotating) is key to mastering this chapter. Good luck with your preparation!