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Geometry - Symmetry: Line of Symmetry and Reflection

Grade 6ICSE

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

🔑Concepts

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Linear Symmetry: A figure is said to have linear symmetry if it can be folded along a line such that the two parts overlap exactly. This line is called the 'Line of Symmetry' or 'Axis of Symmetry'. Visually, imagine a butterfly; its body serves as a vertical line of symmetry where the left wing is a perfect mirror image of the right wing.

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Types of Lines of Symmetry: Lines of symmetry can be vertical, horizontal, or diagonal. For instance, the letter 'M' has one vertical line of symmetry, the letter 'E' has one horizontal line, and a square has four lines of symmetry: one vertical, one horizontal, and two diagonal lines passing through opposite corners.

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Symmetry in Regular Polygons: A regular polygon (a shape with all sides and angles equal) has a specific number of lines of symmetry equal to its number of sides nn. For example, an equilateral triangle has 33 lines of symmetry, a square has 44, and a regular pentagon has 55. Visually, these lines always pass through the center of the polygon.

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Reflectional Symmetry: Reflection is a transformation where a figure is 'flipped' over a mirror line. The resulting image is the same size and shape as the original, but its orientation is reversed. If you place a mirror along the line of symmetry of a heart shape, the reflected half will perfectly complete the figure.

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Distance Property of Reflection: In a reflection, every point on the object and its corresponding point on the image are at the exact same perpendicular distance from the mirror line. For example, if a point PP is 3 cm3 \text{ cm} to the left of a vertical mirror line, its image P′P' will appear exactly 3 cm3 \text{ cm} to the right of that line.

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Lateral Inversion: When an object is reflected in a mirror, the left side appears as the right side and the right side appears as the left side in the image. This is called lateral inversion. For example, if you hold up your left hand to a mirror, the image will appear to be a right hand.

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Symmetry in the English Alphabet: Different letters exhibit different types of symmetry. Letters like 'A', 'W', and 'V' have vertical symmetry; 'B', 'C', and 'D' have horizontal symmetry; 'H', 'I', and 'X' have both; while letters like 'F', 'G', and 'L' have no lines of symmetry.

📐Formulae

Number of lines of symmetry in a regular polygon=n\text{Number of lines of symmetry in a regular polygon} = n

Perpendicular distance of object from mirror line=Perpendicular distance of image from mirror line\text{Perpendicular distance of object from mirror line} = \text{Perpendicular distance of image from mirror line}

Total distance between object and its image=2×distance from mirror line\text{Total distance between object and its image} = 2 \times \text{distance from mirror line}

Reflection of point (x,y) across the x-axis→(x,−y)\text{Reflection of point } (x, y) \text{ across the x-axis} \rightarrow (x, -y)

Reflection of point (x,y) across the y-axis→(−x,y)\text{Reflection of point } (x, y) \text{ across the y-axis} \rightarrow (-x, y)

💡Examples

Problem 1:

Identify the number of lines of symmetry for a rectangle and a circle.

Solution:

  1. For a rectangle: It has 22 lines of symmetry. These are the lines joining the midpoints of the opposite sides (one vertical and one horizontal). Note that the diagonals of a rectangle are NOT lines of symmetry. \ 2. For a circle: It has an infinite number of lines of symmetry. Any line passing through the center of the circle (the diameter) acts as a line of symmetry.

Explanation:

A line of symmetry must divide the shape into two identical halves that coincide when folded. In a rectangle, folding along a diagonal does not align the corners, so only the lines connecting midpoints work. A circle is perfectly round, so any diameter splits it into two equal semicircles.

Problem 2:

A point AA is located 7 cm7 \text{ cm} away from a mirror line LL. If A′A' is the reflected image of AA, calculate the total distance between the original point AA and its image A′A'.

Solution:

Step 1: Note the distance of the object from the mirror line: d=7 cmd = 7 \text{ cm}. \ Step 2: Use the property of reflection that the image A′A' is at the same distance behind the mirror line as the object is in front of it. \ Distance of A′A' from L=7 cmL = 7 \text{ cm}. \ Step 3: Calculate total distance AA′=Distance of A to L+Distance of A′ to LAA' = \text{Distance of } A \text{ to } L + \text{Distance of } A' \text{ to } L. \ AA′=7 cm+7 cm=14 cmAA' = 7 \text{ cm} + 7 \text{ cm} = 14 \text{ cm}.

Explanation:

The mirror line is the perpendicular bisector of the line segment joining the object and its image. Therefore, the total distance is simply double the distance from the object to the mirror.