Rotations and enlargements
A rotation is a turn about a centre
A rotation needs three facts: the centre, the angle and the direction (clockwise or anticlockwise).
A half turn, 180°, needs no direction. About the origin the rules are:
90° clockwise: (x, y) → (y, −x)
90° anticlockwise: (x, y) → (−y, x)
180°: (x, y) → (−x, −y)
About any other centre C, work with the vector from C to the point, turn that vector, and add it back on to C.
- Rotate P(5, 3) 90° clockwise about C(2, 1)
- From C to P: 3 across, 2 up
- Turned 90° clockwise: (3, 2) → (2, −3)
- P′ = (2 + 2, 1 − 3) = (4, −2)
On a paper, tracing paper is allowed. Trace the shape, hold the centre still with your pencil and turn the paper.
An enlargement multiplies distances from a centre
An enlargement with scale factor k and centre C sends every point to k times as far from C, in the same direction.
Every length is multiplied by k. A scale factor between 0 and 1, such as 1/2, makes the shape smaller: it is still called an enlargement.
To find the centre of an enlargement, draw straight lines through matching vertices of the shape and its image and extend them: they meet at the centre. The scale factor is a length on the image divided by the matching length on the original.
The usual slips
- Turning the wrong way. Clockwise is the way the hands of a clock move.
- Enlarging from the origin when the centre is somewhere else. Always measure from the given centre.
- Writing "Rotation 90° about (2, 1)" with no direction, or "Enlargement, scale factor 3" with no centre. Each loses a mark.
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