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Geometry - Symmetry: Line Symmetry and Rotational Symmetry

Grade 7ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Line Symmetry (Reflection Symmetry): A figure is said to have line symmetry if a line can be drawn dividing the figure into two identical parts such that when the figure is folded about that line, the two parts coincide exactly. This line is called the 'Axis of Symmetry'. Visually, imagine a mirror placed along the line; the reflection would complete the original shape.

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Lines of Symmetry in Regular Polygons: A regular polygon (where all sides and angles are equal) has as many lines of symmetry as the number of its sides. For example, an equilateral triangle has 33 lines of symmetry, a square has 44 lines, and a regular pentagon has 55 lines. In a square, these lines include two joining the midpoints of opposite sides and two along the diagonals.

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Rotational Symmetry: A figure has rotational symmetry if, when rotated about a fixed point (the Center of Rotation) by an angle less than 360∘360^\circ, it looks exactly like the original figure. For example, a square rotated by 90∘90^\circ about its center looks unchanged.

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Order of Rotational Symmetry: This represents the number of times a shape fits onto itself during a full 360∘360^\circ rotation. For instance, a rectangle has an order of rotational symmetry of 22 because it looks the same at 180∘180^\circ and 360∘360^\circ. A regular hexagon has an order of 66.

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Angle of Rotation: The minimum angle through which a figure is rotated to coincide with its original position. For any regular polygon with nn sides, the angle of rotation is calculated as 360∘n\frac{360^\circ}{n}. If a figure has an angle of rotation of 120∘120^\circ, its order of symmetry is 33.

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Point Symmetry: A special case of rotational symmetry where a figure looks the same after a rotation of 180∘180^\circ about its center. This is also called 'symmetry about the origin'. Visually, every point on the shape has a matching point at an equal distance from the center in the opposite direction, like the letter 'S' or 'N'.

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Symmetry in English Alphabet: Different letters exhibit different types of symmetry. Letter 'A' has 11 vertical line of symmetry, 'B' has 11 horizontal line, 'H' and 'I' have both vertical and horizontal lines of symmetry as well as rotational symmetry of order 22. Letters like 'O' have infinitely many lines of symmetry and infinite order of rotational symmetry.

📐Formulae

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

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

Number of lines of symmetry in a regular polygon=n (where n is the number of sides)\text{Number of lines of symmetry in a regular polygon} = n \text{ (where } n \text{ is the number of sides)}

Angle of rotation for a regular polygon of n sides=360∘n\text{Angle of rotation for a regular polygon of } n \text{ sides} = \frac{360^\circ}{n}

💡Examples

Problem 1:

A regular polygon has an angle of rotation equal to 72∘72^\circ. Identify the polygon and determine its number of lines of symmetry.

Solution:

Step 1: Use the formula for the order of rotational symmetry: n=360∘Angle of Rotationn = \frac{360^\circ}{\text{Angle of Rotation}} Step 2: Substitute the given value: n=360∘72∘=5n = \frac{360^\circ}{72^\circ} = 5 Step 3: Since the order of rotation is 55 and it is a regular polygon, the number of sides n=5n = 5. A 55-sided regular polygon is a Regular Pentagon. Step 4: For a regular polygon, the number of lines of symmetry equals the number of sides. Therefore, the number of lines of symmetry is 55.

Explanation:

We first find the order of rotational symmetry using the 360∘360^\circ division rule, which tells us the number of sides of the regular polygon. Once the polygon is identified, we apply the property that regular polygons have lines of symmetry equal to their side count.

Problem 2:

Consider a rectangle that is not a square. List its lines of symmetry and its order of rotational symmetry.

Solution:

Step 1: Visualize the lines of symmetry. A rectangle has two lines of symmetry: one horizontal line passing through the midpoints of the shorter sides, and one vertical line passing through the midpoints of the longer sides. Note: The diagonals of a non-square rectangle are NOT lines of symmetry. Step 2: Calculate rotational symmetry.

  • At 90∘90^\circ rotation, the rectangle stands vertically (different from the original).
  • At 180∘180^\circ rotation, it looks identical to the original.
  • At 270∘270^\circ rotation, it stands vertically again.
  • At 360∘360^\circ rotation, it returns to the original position. Step 3: The rectangle looks identical 22 times during a full turn. So, the Order of Rotational Symmetry is 22.

Explanation:

This example distinguishes between a square and a rectangle. While a square has 44 lines of symmetry, a rectangle only has 22. Its rotational symmetry is confirmed by checking how many times it matches its original orientation within a full circle.