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Geometry - Symmetry

Grade 11IGCSE

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

🔑Concepts

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Line Symmetry (Reflective Symmetry): A property where a shape can be folded along a line so that the two halves match exactly.

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Rotational Symmetry: A property where a shape looks the same after a rotation of less than 360° about its center.

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Order of Rotational Symmetry: The number of times a shape looks identical to its original position during a full 360° turn.

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Planes of Symmetry: The 3D equivalent of line symmetry, where a plane divides a solid into two congruent mirror images.

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Symmetry in Regular Polygons: A regular polygon with 'n' sides has 'n' lines of symmetry and rotational symmetry of order 'n'.

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Center of Rotation: The fixed point around which a shape is rotated to demonstrate rotational symmetry.

📐Formulae

Order of Rotational Symmetry (Regular Polygon)=n\text{Order of Rotational Symmetry (Regular Polygon)} = n

Number of Lines of Symmetry (Regular Polygon)=n\text{Number of Lines of Symmetry (Regular Polygon)} = n

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

Reflection in x-axis:(x,y)→(x,−y)\text{Reflection in } x\text{-axis}: (x, y) \rightarrow (x, -y)

Reflection in y-axis:(x,y)→(−x,y)\text{Reflection in } y\text{-axis}: (x, y) \rightarrow (-x, y)

Reflection in line y=x:(x,y)→(y,x)\text{Reflection in line } y = x: (x, y) \rightarrow (y, x)

💡Examples

Problem 1:

Identify the number of lines of symmetry and the order of rotational symmetry for a Rhombus.

Solution:

Lines of symmetry: 2; Order of rotational symmetry: 2.

Explanation:

A rhombus has two lines of symmetry, which are its diagonals. It looks the same twice (at 180° and 360°) during a full rotation, giving it an order of 2.

Problem 2:

A regular hexagon has a side length of 5 cm. State its number of lines of symmetry and the smallest angle it must be rotated by to look identical to its starting position.

Solution:

Lines: 6; Angle: 60°.

Explanation:

For any regular n-gon, there are n lines of symmetry. The angle of rotation is calculated as 360° divided by the order (n), so 360/6=60∘360 / 6 = 60^\circ.

Problem 3:

How many planes of symmetry does a cuboid with dimensions 5cm x 5cm x 10cm have?

Solution:

5 planes of symmetry.

Explanation:

Because two sides are equal (square cross-section), it has: 2 planes passing through the diagonals of the square face, 2 planes bisecting the opposite sides of the square face, and 1 plane bisecting the 10cm length. Total = 5.