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WBB • Class 7 • Mathematics (গণিত প্রভা) • Ch 18

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Symmetry

Welcome to Chapter 18 "Symmetry" of the West Bengal Board of Secondary Education (WBBSE) Class 7 Mathematics (Ganit Prabha) curriculum. This module provides a complete exploration of line symmetry (reflectional symmetry), axes of symmetry, inventory of symmetry lines across triangles, quadrilaterals, regular polygons, and the English alphabet, rotational symmetry mechanics (centre of rotation, angle of rotation, order of symmetry $n = 360^\circ / \theta$), and figures possessing dual symmetry using TargetExams Gold-Standard 5-step pedagogy.

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Overview Core Concepts Formulas & Identities Solved Examples Common Traps Summary Self Check Take MCQ Test

❄️ Snowflakes & Ceiling Fans: How Many Times Does a Shape Replicate Itself in One Turn?

As a 4-blade ceiling fan spins smoothly, how many times in one full 360-degree rotation does it appear completely identical to its starting configuration?

Every time it rotates by just $90^\circ$, the 4 blades match their original positions perfectly—yielding an exact replication 4 times in a single revolution! This mathematical property is known as Rotational Symmetry of Order 4.

Likewise, folding a butterfly or the Taj Mahal along a central vertical axis creates two perfectly overlapping halves—the hallmark of Line Symmetry. Symmetry is nature fundamental aesthetic and structural code.

Why This Chapter Matters

Before You Begin (Prerequisites)

Fundamental properties of 2D shapes (triangles, quadrilaterals, circles)
Angle measurements and complete circular turn ($360^\circ$)
Mirror reflections and folding along crease lines
Basic fraction arithmetic and division

What You Will Learn (Core Objectives)

Distinguish clearly between line symmetry and rotational symmetry
Determine the exact number of symmetry lines for any geometric shape or letter
Calculate rotational order using $n = 360^\circ / \theta$
Explain anomalous cases like parallelograms (0 reflection lines, order 2 rotation)
Analyze figures that possess both reflectional and rotational symmetries

Chapter Roadmap & Progression

1 Concept 1: Line Symmetry & Mirror R...

2 Concept 2: Symmetry Lines Across Ge...

3 Concept 3: Rotational Symmetry (Cen...

4 Concept 4: Dual Symmetry Figures &...

5 Concept 5: Curriculum Synthesis & B...

Complete Concept Guide (100% Curriculum Coverage)

Concept 1: Line Symmetry & Mirror Reflection Fundamentals

Step 1

Definition

A 2D figure has line symmetry if folding it along a line divides it into two congruent halves that coincide exactly.

Step 2

Mirror Line Principle

Placing a mirror along the axis reflects one half to recreate the complete original shape.

Step 3

Example

An isosceles triangle has 1 line of symmetry along the median to the unequal base.

Step 4

Trap

Scalene triangles have 0 lines of symmetry.

Step 5

Nature

Biology: Human bilateral symmetry and butterfly wings are classic reflectional symmetry examples.

Concept 2: Symmetry Lines Across Geometric Shapes & Regular Polygons

Step 1

Triangles

• Scalene: 0 | Isosceles: 1 | Equilateral: 3.

Step 2

Quadrilaterals

• Rectangle: 2 | Rhombus: 2 | Square: 4 | Parallelogram: 0.

Step 3

Regular Polygons & Circle

A regular $n$-gon has exactly $n$ axes of symmetry. Circles have infinitely many axes.

Step 4

Rectangle Diagonal Trap

Diagonals of a non-square rectangle are NOT lines of symmetry.

Step 5

Tessellation

Floor Tessellations: Square and regular hexagonal tiles tessellate gaplessly due to symmetry.

Concept 3: Rotational Symmetry (Centre, Angle & Order)

Step 1

Definitions

When rotated about a fixed centre through angle $\theta < 360^\circ$, if a shape looks identical to its start position, it has rotational symmetry.

Step 2

Order Formula

$$\text{Order of Symmetry} = \frac{360^\circ}{\text{Angle of Rotation}}$$

Step 3

Key Values

• Equilateral triangle: $120^\circ$ (order 3)
• Square: $90^\circ$ (order 4)
• Rectangle: $180^\circ$ (order 2).

Step 4

Caution

Order must always be a positive integer $\ge 2$.

Step 5

Mechanical Engineering

Gears & Turbines: Rotational symmetry balances centrifugal forces in engine turbines.

Concept 4: Dual Symmetry Figures & Alphabet Case Studies

Step 1

Dual Symmetry Shapes

Shapes exhibiting both line reflection and rotation: square, rectangle, rhombus, equilateral triangle, circle.

Step 2

Comparative Matrix

• Square: 4 axes, order 4
• Rectangle: 2 axes, order 2
• Parallelogram: 0 axes, order 2.

Step 3

Alphabet Letter Study

H, I, O, X possess both horizontal and vertical axes plus order 2 rotation. S, N, Z possess order 2 rotation without reflection lines.

Step 4

Trap

Do not confuse letter S as reflectionally symmetric.

Step 5

Logo Aesthetics

Branding: Mercedes-Benz and Audi badges leverage this dual symmetry principle.

Concept 5: Curriculum Synthesis & Board Exam Problems

Step 1

Exam Question Formats

Drawing dashed symmetry axes, computing orders from angles, identifying shapes with rotational but no reflection symmetry.

Step 2

Scoring Checklist

Use dashed lines (---) for axes; always state degrees ($^\circ$) for rotation angles.

Step 3

Model Problem

Isosceles trapezium has 1 reflection line and no rotational symmetry.

Step 4

Caution

Order cannot be a fraction or negative number.

Step 5

Higher Physics

Crystallography: Mineral crystal classifications are governed entirely by spatial symmetry.

Key Formulas, Identities & Theorems

Order of Rotational Symmetry

$$n = \frac{360^\circ}{\theta}$$

Where $\theta$ = smallest angle of rotation and $n$ = order of rotational symmetry.

Regular Polygon Symmetry Rule

$$\text{Lines of Symmetry} = n, \quad \text{Order} = n, \quad \theta = \frac{360^\circ}{n}$$

Holds for any regular polygon with $n$ equal sides and angles.

Rectangle Symmetry Profile

$$\text{Lines of Symmetry} = 2, \quad \text{Order} = 2, \quad \theta = 180^\circ$$

Midpoint bisectors are axes; diagonals are NOT lines of symmetry!

Square Symmetry Profile

$$\text{Lines of Symmetry} = 4, \quad \text{Order} = 4, \quad \theta = 90^\circ$$

2 midpoint bisectors + 2 diagonals.

Parallelogram Anomaly

$$\text{Lines of Symmetry} = 0, \quad \text{Order} = 2, \quad \theta = 180^\circ$$

Possesses order 2 rotational symmetry despite having zero reflection lines.

Circle Infinite Symmetry

$$\text{Lines of Symmetry} = \infty, \quad \text{Order} = \infty$$

Any line passing through the centre (diameter) is an axis of symmetry.

Conceptual Solved Examples & Case Studies

Example 1

State the number of lines of symmetry and the order of rotational symmetry for: (a) Equilateral Triangle, (b) Rectangle, and (c) Regular Hexagon.

Step-by-Step Solution:

(a) Equilateral Triangle: Lines of symmetry $= 3$; Angle $= 120^\circ$; Order $= 360 / 120 = 3$.
(b) Rectangle: Lines of symmetry $= 2$; Angle $= 180^\circ$; Order $= 360 / 180 = 2$.
(c) Regular Hexagon: Lines of symmetry $= 6$; Angle $= 360 / 6 = 60^\circ$; Order $= 6$.

Example 2

Which capital letters of the English alphabet have: (a) only vertical line symmetry, (b) only horizontal line symmetry, and (c) both line and rotational symmetry?

Step-by-Step Solution:

(a) Only vertical symmetry: A, M, T, U, V, W, Y.
(b) Only horizontal symmetry: B, C, D, E, K.
(c) Both horizontal & vertical axes AND $180^\circ$ rotational symmetry: H, I, O, X.
(Note: Letters S, N, Z have no reflection line, but have order 2 rotational symmetry).

Example 3

A geometric figure has an angle of rotation of $45^\circ$. What is its order of rotational symmetry, and what regular polygon could it be?

Step-by-Step Solution:

Order $n = 360^\circ / 45^\circ = 8$. The figure can be a Regular Octagon.

Common Misconceptions & Examiner Traps

Common Misconception

Believing that the diagonal of a rectangle is a line of symmetry.

Scientific Reality & Correction

Folding a non-square rectangle along its diagonal does NOT make the halves coincide. Only squares and rhombuses have diagonal reflection symmetry.

Common Misconception

Assuming that a shape without line symmetry cannot have rotational symmetry.

Scientific Reality & Correction

Parallelograms and letters S, N, Z have 0 reflection lines, yet possess order 2 rotational symmetry at $180^\circ$.

Common Misconception

Claiming an order of 1 at 360° represents rotational symmetry.

Scientific Reality & Correction

Every object looks identical after a full 360° turn. Mathematical rotational symmetry requires an angle less than 360° and an order $\ge 2$.

Line Symmetry Axes vs Rotational Symmetry Orders Model

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All Class 7 Mathematics (গণিত প্রভা) Chapters

Ch 1: Revision of Previous Lessons Ch 2: Ratio Ch 3: Proportion Ch 4: Addition, Subtraction, Multiplication and Division of Integers Ch 5: Concept of Indices Ch 6: Algebraic Operations Ch 7: Construction of Specific Angles with Compass Ch 8: Construction of Triangles Ch 9: Concept of Congruence Ch 10: Approximation Ch 11: Square Root of Fractions Ch 12: Algebraic Formulae Ch 13: Parallel Lines and Transversals Ch 14: Properties of Triangles Ch 15: Time and Distance Ch 16: Double Bar Graph Ch 17: Area of Rectangle and Square Ch 18: Symmetry Ch 19: Factorization Ch 20: Classification of Quadrilaterals Ch 21: Construction of Quadrilaterals Ch 22: Formation and Solution of Equations

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