krit.club logo

Geometry - Symmetry: Line Symmetry and Rotational Symmetry

Grade 7ICSE

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

🔑Concepts

•

Line Symmetry (Reflectional Symmetry): A figure has line symmetry if a line can be drawn dividing the figure into two identical parts that are mirror images of each other. This line is called the axis of symmetry.

Isosceles triangle with a vertical line of symmetry passing through the top vertex.
•

Rotational Symmetry: A figure has rotational symmetry if it looks exactly the same more than once during a full 360∘360^\circ turn around its center. The fixed point is called the Center of Rotation.

A rectangle showing a center of rotation at the intersection of its diagonals.
•

Order of Rotational Symmetry: This is the number of times a figure fits onto itself in one complete rotation of 360∘360^\circ. For example, a square has an order of 4.

A square illustrating 4 positions of symmetry during rotation.
•

Regular Polygons: A regular polygon with nn sides has nn lines of symmetry and an order of rotational symmetry equal to nn. Each interior angle of rotation is 360∘n\frac{360^\circ}{n}.

A regular pentagon with one of its 5 lines of symmetry shown.

📐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.

Problem 3:

Examine the letter 'H'. State the number of lines of symmetry and the order of rotational symmetry.

Letter H with horizontal and vertical axes of symmetry.

Solution:

  1. Lines of Symmetry: The letter 'H' can be folded horizontally and vertically to match perfectly. So, it has 2 lines of symmetry.
  2. Rotational Symmetry: When rotated by 180∘180^\circ and 360∘360^\circ, it looks the same. The order of rotational symmetry is 2.

Explanation:

Objects like 'H' possess both vertical and horizontal reflectional symmetry. Since it fits onto itself twice in a full circle, the order is 360∘/180∘=2360^\circ / 180^\circ = 2.

Problem 4:

An equilateral triangle is rotated about its center. Determine the minimum angle of rotation required for the triangle to occupy the same position, and find its order of rotational symmetry.

Equilateral triangle showing rotation of 120 degrees around the center.

Solution:

  1. Order of Rotational Symmetry: A regular polygon with nn sides has an order of nn. For an equilateral triangle, n=3n = 3.
  2. Angle of Rotation: Angle=360∘n=360∘3=120∘\text{Angle} = \frac{360^\circ}{n} = \frac{360^\circ}{3} = 120^\circ.

Explanation:

In an equilateral triangle, rotating by 120∘120^\circ moves each vertex to the position of the next vertex, making the shape look unchanged.