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Geometry - Symmetry and Transformations (Translation, Rotation, Reflection, Enlargement)

Grade 8IGCSE

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

🔑Concepts

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Line Symmetry: A shape has line symmetry if it can be folded along a line so that the two halves match exactly.

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Rotational Symmetry: The number of times a shape looks the same during a full 360-degree turn (Order of Symmetry).

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Translation: Moving a shape without rotating or flipping it, defined by a column vector.

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Reflection: Flipping a shape over a mirror line; every point is the same distance from the line as its image.

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Rotation: Turning a shape around a fixed point (center) by a specific angle and direction (clockwise or anti-clockwise).

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Enlargement: Changing the size of a shape by a scale factor 'k' from a center of enlargement.

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Congruency: Translation, Reflection, and Rotation produce congruent shapes (same size and shape).

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Similarity: Enlargement produces similar shapes (same shape, different size).

📐Formulae

Translation Vector: (xy) (where x=horizontal shift, y=vertical shift)\text{Translation Vector: } \begin{pmatrix} x \\ y \end{pmatrix} \text{ (where } x = \text{horizontal shift, } y = \text{vertical shift)}

Scale Factor (k)=Length of Image sideLength of Object side\text{Scale Factor (k)} = \frac{\text{Length of Image side}}{\text{Length of Object side}}

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

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

Rotation 180∘ about origin: (x,y)→(−x,−y)\text{Rotation } 180^\circ \text{ about origin: } (x, y) \rightarrow (-x, -y)

Distance from center to image=k×Distance from center to object\text{Distance from center to image} = k \times \text{Distance from center to object}

💡Examples

Problem 1:

Translate the point A(2,3)A(2, 3) by the vector (−45)\begin{pmatrix} -4 \\ 5 \end{pmatrix}. Find the coordinates of the image A′A'.

Solution:

A′=(−2,8)A' = (-2, 8)

Explanation:

Add the xx-component of the vector to the xx-coordinate: 2+(−4)=−22 + (-4) = -2. Add the yy-component to the yy-coordinate: 3+5=83 + 5 = 8.

Problem 2:

A square with side length 5 cm is enlarged by a scale factor of 3. What is the side length and area of the new square?

Solution:

Side length = 15 cm; Area = 225 cm²

Explanation:

The new side length is 5×3=155 \times 3 = 15 cm. The area of the new square is 152=22515^2 = 225 cm² (or original area 25×k225 \times k^2).

Problem 3:

Describe the single transformation that maps the point (3,4)(3, 4) onto (4,3)(4, 3).

Solution:

Reflection in the line y=xy = x.

Explanation:

When the xx and yy coordinates are swapped, it indicates a reflection across the diagonal line where yy equals xx.

Problem 4:

What is the order of rotational symmetry for a regular hexagon?

Solution:

6

Explanation:

A regular hexagon can be rotated by 60∘60^\circ (360/6) six times within a full circle and look identical each time.