Rotation, Reflection, and Translation: KS3 Maths

The three main transformations at KS3 are rotation, reflection, and translation. Rotation turns a shape about a centre point; reflection flips it across a mirror line; translation slides it by a vector. Each must be described fully — a partial description earns partial or no marks in an exam.

What information do you need to describe each transformation?

Every transformation has a minimum set of information required for a complete description. Missing any element means the description is incomplete.

Transformation | Must state | Example
Rotation | Angle, direction (clockwise or anticlockwise), centre of rotation | Rotation 90° clockwise about (0, 0)
Reflection | Equation of the mirror line | Reflection in the line y = x
Translation | Column vector | Translation by vector (3, −2)
Enlargement | Scale factor, centre of enlargement | Enlargement scale factor 2, centre (1, 1)

Enlargement is listed for completeness — it is covered separately. Focus here is on the three main transformations.

How do you describe and perform a rotation?

A rotation turns a shape through a given angle about a fixed point called the centre of rotation. To describe a rotation fully you need three things: the angle, the direction, and the centre.

Common rotations and their coordinate rules (rotating about the origin):

Rotation | Rule for point (x, y) | Result
90° clockwise | (x, y) → (y, −x) | Rotate quarter turn right
90° anticlockwise | (x, y) → (−y, x) | Rotate quarter turn left
180° (either direction) | (x, y) → (−x, −y) | Half turn

Worked example: Rotate triangle A with vertices (1, 1), (3, 1), (3, 4) by 90° clockwise about the origin.

Apply the rule (x, y) → (y, −x) to each vertex:

(1, 1) → (1, −1)

(3, 1) → (1, −3)

(3, 4) → (4, −3)

Plot the image with vertices (1, −1), (1, −3), (4, −3) and label it A′.

When the centre is not the origin, translate so the centre becomes the origin, apply the rotation rule, then translate back.

How do you describe and perform a reflection?

A reflection flips a shape across a straight line called the mirror line. Every point on the original shape maps to a point the same perpendicular distance from the mirror line, on the opposite side.

Common mirror lines and their effects:

Mirror line | Effect on point (x, y)
x-axis (y = 0) | (x, y) → (x, −y)
y-axis (x = 0) | (x, y) → (−x, y)
y = x | (x, y) → (y, x)
y = −x | (x, y) → (−y, −x)

Worked example: Reflect triangle B with vertices (2, 1), (5, 1), (5, 4) in the line y = x.

Apply the rule (x, y) → (y, x) to each vertex:

(2, 1) → (1, 2)

(5, 1) → (1, 5)

(5, 4) → (4, 5)

Plot the image with vertices (1, 2), (1, 5), (4, 5) and label it B′.

When the mirror line is not one of the four standard lines, draw the perpendicular from each vertex to the mirror line, extend the same distance on the other side, and mark the image point there.

How do you describe and perform a translation?

A translation slides every point of a shape by the same amount in the same direction, with no rotation or reflection. The movement is described by a column vector: the top number gives horizontal movement (positive = right) and the bottom number gives vertical movement (positive = up).

Worked example: Translate shape C with vertices (−1, 2), (−1, 5), (2, 5), (2, 2) by the vector (4, −3).

Add 4 to every x-coordinate and −3 to every y-coordinate:

(−1, 2) → (3, −1)

(−1, 5) → (3, 2)

(2, 5) → (6, 2)

(2, 2) → (6, −1)

Plot the image with vertices (3, −1), (3, 2), (6, 2), (6, −1) and label it C′.

To describe a translation from a diagram, pick any vertex, count how far it has moved right/left and up/down, and write those values as a column vector.

How do you find the centre of rotation?

When a question gives you a shape and its rotated image and asks for the centre of rotation, use the following method:

Join a vertex of the original shape to the corresponding vertex of the image. Draw the perpendicular bisector of that line segment.

Repeat with a different pair of corresponding vertices.

The two perpendicular bisectors intersect at the centre of rotation.

Verify by checking that a third pair of corresponding vertices is consistent with that centre.

You can also use tracing paper in an exam: place the tracing paper over the original, trace the shape, then try different centre points while rotating the tracing paper until the traced shape overlaps the image. Mark the pinhole — that is the centre.

What properties does each transformation preserve?

All three transformations are isometries — they preserve the size and shape of the object.

Property | Rotation | Reflection | Translation
Side lengths preserved? | Yes | Yes | Yes
Angles preserved? | Yes | Yes | Yes
Orientation (handedness) preserved? | Yes | No | Yes
Shape congruent to original? | Yes | Yes | Yes

The key distinction is orientation. After a reflection, the shape is "mirror-reversed" — a clock face that reads clockwise will read anticlockwise in its mirror image. After a rotation or translation, the orientation is unchanged. This is why reflections are sometimes called indirect congruences and rotations/translations are direct congruences.

Frequently asked questions

What is the difference between clockwise and anticlockwise?

Clockwise follows the direction of clock hands: top → right → bottom → left. Anticlockwise is the opposite: top → left → bottom → right. In a GCSE answer, you must always state which direction when describing a rotation. A 90° clockwise rotation is not the same transformation as a 90° anticlockwise rotation — they produce different image positions. The exception is 180°, which gives the same result in either direction.

How do I write the equation of a mirror line from a diagram?

Horizontal mirror lines have equations of the form y = k (where k is a constant). Vertical mirror lines have equations of the form x = k. Diagonal mirror lines at 45° are y = x or y = −x. If the mirror line does not pass through the origin, use one point on the line to find k. For example, a horizontal line through (0, 3) is y = 3. Stating "the x-axis" or "the y-axis" instead of the equation is less precise and may not score full marks.

Can I use tracing paper in a GCSE exam?

Yes — tracing paper is permitted and useful in many transformation questions. It is particularly helpful when finding the centre of rotation or when performing a rotation where the centre is not the origin. Draw the shape on tracing paper, hold a pencil at the trial centre, and rotate until the shape lines up with the image. When it does, the pencil tip marks the centre. Tracing paper can also help you visualise a reflection or check whether a translation vector is correct.

Why does a reflection reverse orientation but rotation and translation do not?

A reflection involves flipping a shape through a line, which reverses the "handedness" of the shape — left becomes right. Imagine a letter R reflected in a vertical mirror: it becomes a reversed R. Rotation and translation move the shape without any flipping, so left stays left. This distinction is important in geometry proofs: if a shape and its image are congruent but have opposite orientation, at least one reflection must be involved in the transformation.

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