Transformations and Vectors

A transformation changes the position, size or orientation of a shape. The original shape is the object; the result is the image. At KS3 you study four transformations: reflection, rotation, translation and enlargement.

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Reflection

A reflection produces a mirror image of a shape in a mirror line. Each point is the same perpendicular distance from the mirror line as its image.

Properties: shape and size preserved (congruent); orientation is reversed.

Common mirror lines and their effects on coordinates:

Mirror line	Effect on (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)
x = a	reflect in a vertical line
y = b	reflect in a horizontal line

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Rotation

A rotation turns a shape through an angle (in degrees) about a fixed centre of rotation, in a stated direction (clockwise or anti-clockwise).

Properties: shape and size preserved (congruent); orientation preserved (not flipped).

Common rotations about the origin:

Rotation	Effect on (x, y)
90° clockwise	(x, y) → (y, −x)
90° anti-clockwise	(x, y) → (−y, x)
180°	(x, y) → (−x, −y)

To find the centre of rotation: draw a line between a point and its image; the centre lies on the perpendicular bisector. Repeat for another pair of points — the centre is where the bisectors meet.

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Translation

A translation moves every point the same distance in the same direction, described by a column vector written as (a, b) where a = horizontal shift and b = vertical shift. Positive a = right; positive b = up.

Properties: shape, size and orientation all preserved.

Example: Translate point (3, 4) by vector (5, −2): New position: (3 + 5, 4 + (−2)) = (8, 2)

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Enlargement

An enlargement changes the size of a shape by a scale factor (k) from a fixed centre of enlargement.