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Geometry & measures · Transformations

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Translations & enlargements

Column vectors as translations, enlargement by a positive, fractional or negative scale factor from a centre, finding the centre and scale factor from a picture, and why lengths scale but angles don't.

Geometry & measures · Transformations

Translations & enlargements

Column vectors as translations, enlargement by a positive, fractional or negative scale factor from a centre, finding the centre and scale factor from a picture, and why lengths scale but angles don't.

Why it works

A translation slides every point by the same column vector.

(ab)means a across (right if positive), b up (up if positive).\begin{pmatrix} a \\ b \end{pmatrix} \quad\text{means } a \text{ across (right if positive), } b \text{ up (up if positive).}

Top number across, bottom number up — always. Writing the pair as coordinates (a,b)(a, b) instead of a column vector loses the mark, and so does swapping the two numbers.

An enlargement scales distance from the centre. For scale factor kk about centre CC, every point PP moves along the ray CPCP to the point kk times as far from CC:

CP′=k×CP.CP' = k \times CP.

That's why the centre matters as much as the factor — the same kk about a different centre puts the image somewhere else. In practice: count the steps from the centre to a vertex, multiply both the across-step and the up-step by kk, and that's the image vertex. Repeat for each vertex.

Lengths multiply by kk; angles do not change. An enlargement is a similarity, so the image is the same shape — corresponding angles are identical and corresponding sides are all in the ratio kk. (Area, though, scales by k2k^2 — that belongs with similar shapes.)

A "fractional" enlargement makes it smaller — and is still called an enlargement. Scale factor 12\frac{1}{2} halves every distance from the centre. In GCSE language it is still an enlargement, never a "reduction", and the scale factor is 12\frac{1}{2}, not 22.

A negative scale factor turns the shape through the centre. For k=−2k = -2, go from the centre to the vertex, then travel backwards through the centre to twice the distance on the opposite side. The image ends up upside-down (rotated 180°180°) as well as twice as big.

Reading a transformation off a picture. To find the scale factor, divide an image length by the matching object length. To find the centre, draw a straight line through each pair of corresponding vertices and extend them — all the rays meet at the centre.

Describing translations and enlargements — the checklist again:
  • Translation → the column vector, written as a column vector.
  • Enlargement → the scale factor and the centre of enlargement.