TransformationsAQA GCSE Maths: Revision notes

The four main transformations are reflection, rotation, translation, and enlargement. Each transforms a shape in a different way.

Reflection is a mirror image of a shape across a line (the mirror line or line of symmetry). To describe a reflection, you must state the equation of the mirror line (e.g. y = x, x = 2, y = 0).

Rotation turns a shape about a fixed point (the centre of rotation). To describe a rotation completely, you must state:

1. The centre of rotation (coordinates)
2. The angle of rotation (in degrees)
3. The direction (clockwise or anticlockwise)

Translation moves a shape without rotating or reflecting it. Describe a translation using a column vector in the form $\begin{pmatrix} a \\ b \end{pmatrix}$, where a is the horizontal movement (right is positive) and b is the vertical movement (up is positive).

Enlargement changes the size of a shape. To describe an enlargement, state the scale factor and the centre of enlargement. A scale factor greater than 1 makes the shape bigger; between 0 and 1 makes it smaller.

To perform a reflection, follow these steps:

1. Identify the mirror line (e.g. x = 2, y = x, the x-axis, the y-axis)
2. For each vertex of the shape, measure its perpendicular distance to the mirror line
3. Plot the image vertex the same distance on the opposite side of the mirror line
4. Join the image vertices to form the reflected shape

Common mirror lines and how to find image coordinates:

To identify a reflection from a diagram, find the mirror line by looking for the perpendicular bisector of the line segment joining any point on the original shape to its image.

To perform a rotation, follow these steps:

1. Identify the centre of rotation (given as coordinates, often the origin)
2. Identify the angle (e.g. 90°, 180°, 270°) and direction (clockwise or anticlockwise)
3. Using tracing paper or a protractor: place a pin at the centre of rotation, rotate the shape through the specified angle in the specified direction
4. Mark the new positions of each vertex and join them to form the image

Key rotation facts:

• A 180° rotation is the same clockwise or anticlockwise
• Common rotations: 90° clockwise = 270° anticlockwise (and vice versa)
• The centre of rotation is the only point that does not move
• All distances from the centre of rotation to vertices remain the same
• If no direction is specified but the angle is 180°, state either clockwise or anticlockwise

For rotations about the origin on a coordinate grid:

• 90° clockwise: point (x, y) → (y, −x)
• 90° anticlockwise: point (x, y) → (−y, x)
• 180°: point (x, y) → (−x, −y)
• 270° clockwise (= 90° anticlockwise): point (x, y) → (−y, x)

To identify a rotation, find the centre by constructing the perpendicular bisectors of line segments joining corresponding points on the original and image shapes.

Enlargement is a transformation that changes the size of a shape. Every length in the shape is multiplied by the scale factor from a centre of enlargement.

To perform an enlargement:

1. Identify the centre of enlargement and scale factor
2. Measure the distance from the centre to each vertex of the original shape
3. Multiply each distance by the scale factor
4. Plot the image vertex along the same direction from the centre
5. Join the image vertices

Scale factor effects:

• Scale factor > 1: image is larger than the original
• 0 < scale factor < 1 (fractional): image is smaller than the original
• Scale factor = 1: no change (identity transformation)
• Negative scale factor: image is on the opposite side of the centre and inverted

Negative scale factors (Higher Tier only): A negative scale factor of −2, for example, means the image is twice as large and on the opposite side of the centre of enlargement. To find an image point: if the original point is at distance d from the centre in direction $\theta$, the image is at distance 2d from the centre in direction $\theta + 180°$.

To find the scale factor from original and image: divide any length in the image by the corresponding length in the original.

To find the centre of enlargement: draw a straight line through corresponding points on the original and image shapes; these lines meet at the centre.

Congruent shapes are identical in shape and size. They are the same shape after a reflection, rotation, or translation. Congruent triangles have corresponding sides equal in length and corresponding angles equal.

Four conditions for triangle congruence (always state which one):

Similar shapes have the same shape but different sizes. All corresponding angles are equal and all corresponding side lengths are in the same ratio (the scale factor).

Key difference: congruent shapes are identical (scale factor = 1); similar shapes are enlarged or reduced versions of each other (any positive scale factor).

To prove shapes are similar, show that:

• All corresponding angles are equal, OR
• All corresponding side lengths are in the same ratio

When two shapes are similar with a linear scale factor of k:

• Linear measurements (lengths, widths, heights, perimeters) scale by factor k
• Areas scale by factor k² (the scale factor squared)
• Volumes scale by factor k³ (the scale factor cubed)

Working with areas:

If the linear scale factor is 2, the area scale factor is 2² = 4. So if the original shape has area 10 cm², the enlarged shape has area 10 × 4 = 40 cm².

Working with volumes:

If the linear scale factor is 3, the volume scale factor is 3³ = 27. So if the original solid has volume 5 cm³, the enlarged solid has volume 5 × 27 = 135 cm³.

To find the linear scale factor from areas: if the area scale factor is 4, the linear scale factor is √4 = 2.

To find the linear scale factor from volumes: if the volume scale factor is 8, the linear scale factor is ∛8 = 2.

Finding unknown lengths in similar shapes:

1. Identify two corresponding lengths (one known in each shape)
2. Calculate the linear scale factor: $k = \frac{\text{length in image}}{\text{corresponding length in original}}$
3. Multiply any unknown length in the original by k to find the corresponding length in the image (or divide if finding the original)

Example calculation: Triangle A has sides 3, 4, 5 cm. Triangle B is similar with a corresponding side of 6 cm. Scale factor = 6 ÷ 3 = 2. The other sides of Triangle B are 4 × 2 = 8 cm and 5 × 2 = 10 cm.

Combined transformations involve applying two or more transformations in sequence. The result of the first transformation becomes the input for the second.

How to apply combined transformations:

1. Perform the first transformation on the original shape and note the vertices of the intermediate image
2. Perform the second transformation on the intermediate image
3. State the overall combined transformation if possible (sometimes two transformations can be described as a single equivalent transformation)

Key combinations:

• Two reflections in parallel mirror lines = one translation (perpendicular distance = twice the distance between the lines)
• Two reflections in intersecting mirror lines = one rotation about the intersection point (angle = twice the angle between the lines)
• A reflection followed by a translation = a glide reflection
• A reflection and a rotation in certain orders may combine to give a single transformation

On a coordinate grid:

1. Apply transformations one at a time to each vertex
2. Check by looking for invariant points (points that do not move)
3. Always draw the intermediate shapes to show your working

Example: Reflect triangle ABC in the y-axis, then translate by $\begin{pmatrix} 2 \\ -1 \end{pmatrix}$. First, reflect each point across y = 0; then shift each reflected point right 2 and down 1.

Finding the single equivalent transformation can save time: identify the overall effect on a general point and describe it in a single transformation.