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Does it Look the Same? - Symmetry and Mirror Halves

Grade 5CBSE

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 shape or a pattern into two identical mirror halves. If you fold the shape along this line, the two halves will overlap perfectly. For example, a heart shape has one vertical line of symmetry down the middle, making the left and right sides mirror images.

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Mirror reflection occurs when one half of a figure is the exact reflection of the other. You can test this by placing a mirror on the line of symmetry; the reflection in the mirror should complete the shape so it looks exactly like the original. This is common in symmetrical letters like 'A', 'M', and 'U'.

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A 12\frac{1}{2} turn (or half-turn) means rotating a shape by 180∘180^{\circ} around a center point. A shape has half-turn symmetry if it looks exactly the same after being turned upside down. For instance, the number 88 and the letter 'S' look identical after a 12\frac{1}{2} turn.

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A 14\frac{1}{4} turn (or quarter-turn) involves rotating a shape by 90∘90^{\circ}. A square looks exactly the same after a 14\frac{1}{4} turn because all its sides and angles are equal. However, a rectangle only looks the same after a 12\frac{1}{2} turn, not a 14\frac{1}{4} turn.

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Shapes can also have 13\frac{1}{3} turn (120∘120^{\circ}) and 16\frac{1}{6} turn (60∘60^{\circ}) symmetry. An equilateral triangle, which has three equal sides, looks the same after a 13\frac{1}{3} turn. A regular hexagon or a star with six points will look the same after a 16\frac{1}{6} turn.

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Symmetry in the alphabet can be horizontal, vertical, or both. Letters like 'E', 'B', and 'C' have a horizontal line of symmetry (splitting the top and bottom), while letters like 'W', 'Y', and 'V' have a vertical line of symmetry (splitting the left and right). Letters like 'H', 'I', 'O', and 'X' have both vertical and horizontal lines of symmetry.

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Patterns and tiles often use symmetry to create repeating designs. In a symmetrical pattern, the unit shape is reflected or rotated repeatedly to fill a space. To check for symmetry in a pattern, look for a central point or line where the design repeats itself in a mirrored fashion.

📐Formulae

Angle of Turn=Fraction of Turn×360∘\text{Angle of Turn} = \text{Fraction of Turn} \times 360^{\circ}

12 turn=180∘\frac{1}{2} \text{ turn} = 180^{\circ}

14 turn=90∘\frac{1}{4} \text{ turn} = 90^{\circ}

13 turn=120∘\frac{1}{3} \text{ turn} = 120^{\circ}

16 turn=60∘\frac{1}{6} \text{ turn} = 60^{\circ}

Number of Lines of Symmetry in a Regular Polygon=Number of Sides\text{Number of Lines of Symmetry in a Regular Polygon} = \text{Number of Sides}

💡Examples

Problem 1:

Which of the following letters looks the same after a 12\frac{1}{2} turn: H, L, P, Z?

Solution:

  1. Analyze letter 'H': Turning it 180∘180^{\circ} results in the same shape.
  2. Analyze letter 'L': Turning it 180∘180^{\circ} makes it look like an upside-down 'L', which is different.
  3. Analyze letter 'P': Turning it 180∘180^{\circ} makes it look like a 'd', which is different.
  4. Analyze letter 'Z': Turning it 180∘180^{\circ} results in the same 'Z' shape. Final Answer: H and Z.

Explanation:

Rotational symmetry of 12\frac{1}{2} turn means the figure must look identical when rotated halfway around a circle (180∘180^{\circ}).

Problem 2:

How many lines of symmetry does a regular hexagon have?

Solution:

  1. A regular hexagon has 66 equal sides and 66 equal angles.
  2. You can draw lines through the opposite vertices (corners): there are 33 such lines.
  3. You can draw lines through the midpoints of opposite sides: there are 33 such lines.
  4. Total lines = 3+3=63 + 3 = 6.

Explanation:

For any regular polygon (where all sides and angles are equal), the number of lines of symmetry is always equal to the number of its sides (nn).