Geometry & measures
Transformations
Year 7 · Year 8 · Year 9
- ✓Identify and describe reflections, rotations, translations, and enlargements.
- ✓Perform reflections of 2D shapes in horizontal, vertical, and diagonal mirror lines.
- ✓Perform rotations of 2D shapes about a given centre of rotation through multiples of 90 degrees.
- ✓Perform translations of 2D shapes using a given column vector.
- ✓Perform enlargements of 2D shapes from a given centre with a positive integer scale factor.
Key concepts
A reflection 'flips' a shape over a line, called the mirror line. The reflected image is congruent to the original shape, but it is a mirror image. Each point on the original shape is the same perpendicular distance from the mirror line as its corresponding point on the reflected image.
A rotation 'turns' a shape around a fixed point, called the centre of rotation. To describe a rotation fully, you need the centre of rotation, the angle of rotation (e.g., 90°, 180°, 270°), and the direction of rotation (clockwise or anti-clockwise). The rotated image is congruent to the original shape.
A translation 'slides' a shape from one position to another without changing its orientation or size. It is described by a column vector, which indicates how many units the shape moves horizontally (top number) and vertically (bottom number). A positive number means right or up, a negative number means left or down.
An enlargement changes the size of a shape by a given scale factor from a fixed point, called the centre of enlargement. The enlarged image is similar to the original shape (angles are preserved, but side lengths and area change). A scale factor greater than 1 makes the shape bigger, while a scale factor between 0 and 1 makes it smaller.
Key facts to remember
- 1Isometries (reflection, rotation, translation) are transformations that preserve the size and shape of the object. The image is congruent to the object.
- 2Enlargement changes the size of the object but preserves its shape (angles remain the same). The image is similar to the object.
- 3A reflection requires a mirror line (e.g., x=k, y=k, y=x, y=-x).
- 4A rotation requires a centre of rotation, an angle (e.g., 90°, 180°, 270°), and a direction (clockwise or anti-clockwise).
- 5A translation requires a column vector (horizontal movement, vertical movement).
- 6An enlargement requires a centre of enlargement and a scale factor.
- 7The original shape is called the object, and the transformed shape is called the image.
Worked examples
Example 1
Reflect the triangle A(1,2), B(3,2), C(2,4) in the line y = x.
Answer
The reflected triangle has vertices A'(2,1), B'(2,3), C'(4,2).
When reflecting in the line y=x, the x and y coordinates swap.
Example 2
Rotate the quadrilateral P(1,1), Q(3,1), R(3,3), S(1,3) 90 degrees anti-clockwise about the origin (0,0).
Answer
The rotated quadrilateral has vertices P'(-1,1), Q'(-1,3), R'(-3,3), S'(-3,1).
For a 90-degree anti-clockwise rotation about the origin, a point (x,y) maps to (-y,x).
Example 3
Enlarge the triangle D(2,1), E(4,1), F(2,3) by a scale factor of 2 from the centre of enlargement (1,1).
Answer
The enlarged triangle has vertices D'(3,1), E'(7,1), F'(3,5).
You can also draw lines from the centre of enlargement through each vertex of the object and extend them. Then measure the distance from the centre to each vertex and multiply by the scale factor to find the new distance for the image vertex.
Common mistakes
- ✗Confusing directions for rotation: Mixing up clockwise and anti-clockwise, or rotating in the wrong direction.
- ✗Incorrectly identifying the centre: Using the origin instead of the specified centre for rotations or enlargements.
- ✗Counting squares inaccurately: Errors in counting distances for reflections or vector components for translations.
- ✗Reflecting in the wrong line: Forgetting whether x=k is a vertical line or y=k is a horizontal line, or confusing y=x with y=-x.
- ✗Applying scale factor from the origin: For enlargements, multiplying the object's coordinates by the scale factor instead of finding the vector from the centre of enlargement and then scaling it.
Exam tips
- ★Use tracing paper: This is invaluable for rotations and reflections, allowing you to physically turn or flip the shape to find its image.
- ★Label points clearly: Label the vertices of your object (e.g., A, B, C) and their corresponding image points (A', B', C') to avoid confusion.
- ★Draw construction lines for enlargements: Draw faint lines from the centre of enlargement through each vertex of the object. The image vertices will lie on these lines.
- ★Fully describe transformations: When asked to describe a transformation, ensure you include all necessary information (e.g., 'Rotation, 90 degrees anti-clockwise, about the origin (0,0)').
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