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Symmetry - Lines of Symmetry for Regular Polygons

Grade 7CBSE

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

🔑Concepts

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A line of symmetry is an imaginary line that divides a figure into two identical parts such that when the figure is folded along this line, the two parts coincide or overlap exactly. It acts like a mirror line, where one half is the reflection of the other.

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A regular polygon is a closed geometric figure where all sides have equal length and all interior angles are equal in measure. Common examples include equilateral triangles, squares, and regular pentagons.

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The number of lines of symmetry in any regular polygon is exactly equal to the number of its sides (nn). For example, a regular octagon with 88 sides will have exactly 88 lines of symmetry.

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In a regular polygon with an odd number of sides, such as an equilateral triangle or a regular pentagon, every line of symmetry passes through one vertex and the midpoint of the opposite side. Imagine a line starting from the top point of a pentagon and cutting straight down through the center of the base.

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In a regular polygon with an even number of sides, such as a square or a regular hexagon, lines of symmetry are of two types: some pass through opposite vertices (diagonals), and others pass through the midpoints of opposite sides. In a square, you can visualize two lines forming a '+' shape through the centers and two lines forming an 'X' shape through the corners.

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A regular hexagon possesses 66 lines of symmetry. You can visualize these as three lines connecting opposite corners (vertices) and three lines connecting the centers of opposite flat edges. All these lines intersect at the exact center of the hexagon.

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A circle can be considered a limiting case of a regular polygon with an infinite number of sides. Consequently, it has an infinite number of lines of symmetry, as any straight line passing through its center point will divide it into two identical semicircles.

📐Formulae

Number of lines of symmetry in a regular polygon = nn, where nn is the number of sides.

Interior angle of a regular polygon = (n−2)×180∘n\frac{(n-2) \times 180^{\circ}}{n}

Each line of symmetry in a regular polygon passes through the center of the polygon.

💡Examples

Problem 1:

How many lines of symmetry does a regular pentagon have? Describe their positions.

Solution:

Step 1: Identify the type of polygon. A regular pentagon has n=5n = 5 sides. Step 2: Apply the rule for regular polygons. The number of lines of symmetry equals the number of sides. Step 3: Therefore, a regular pentagon has 55 lines of symmetry. Step 4: Since 55 is an odd number, each line of symmetry passes through one vertex and the midpoint of the side opposite to that vertex.

Explanation:

For any regular polygon with nn sides, the count of lines of symmetry is nn. Since all sides and angles are equal, the shape is perfectly balanced, allowing for 55 distinct fold lines.

Problem 2:

A regular polygon has 1010 lines of symmetry. Name the polygon and calculate its interior angle.

Solution:

Step 1: Determine the number of sides. Since the number of lines of symmetry in a regular polygon is n=10n = 10, the polygon has 1010 sides. A 1010-sided polygon is called a regular decagon. Step 2: Use the formula for the interior angle: Interior Angle=(n−2)×180∘nInterior\ Angle = \frac{(n-2) \times 180^{\circ}}{n}. Step 3: Substitute n=10n = 10: Interior Angle=(10−2)×180∘10Interior\ Angle = \frac{(10-2) \times 180^{\circ}}{10}. Step 4: Calculate: Interior Angle=8×180∘10=8×18∘=144∘Interior\ Angle = \frac{8 \times 180^{\circ}}{10} = 8 \times 18^{\circ} = 144^{\circ}.

Explanation:

This problem links the concept of symmetry directly to the physical properties of the polygon. Once nn is identified from the symmetry lines, standard geometric formulas can be applied.