Congruent and similar shapes; scale factors
Foundation candidates use whole-number scale factors; Intermediate and Higher meet fractional and negative scale factors.
Congruent shapes
Two shapes are CONGRUENT if they have identical size and shape — one can be mapped onto the other by translation, rotation or reflection (no enlargement).
The four standard tests for congruent triangles:
- SSS — three sides equal
- SAS — two sides and the included angle equal
- ASA — two angles and the included side equal
- RHS — right angle, hypotenuse, one other side equal
WJEC Higher will ask: "State, with reasons, why triangles ABC and DEF are congruent." Quote the test name and tick off the matching pairs.
Similar shapes
Two shapes are SIMILAR if their angles match and corresponding sides are in the same ratio. Every enlargement produces a similar shape.
Scale factor (linear) = (new length) ÷ (corresponding original length).
If the scale factor is k:
- Lengths multiply by k.
- Areas multiply by k².
- Volumes multiply by k³.
Fractional scale factors
A scale factor of 1/2 produces an image half the size — it is still an enlargement in the mathematical sense, just a reduction visually. The image lies between the centre of enlargement and the original shape.
Negative scale factors
A scale factor of −2 means: enlarge by 2 AND rotate 180° about the centre of enlargement. The image appears on the OPPOSITE side of the centre. Distances multiply by |k|; orientation flips.
Constructing an enlargement
- Mark centre of enlargement O.
- Measure vector from O to a vertex, multiply by k (with sign).
- Plot new vertex. Repeat for each vertex.
WJEC exam tip
When asked "describe fully", a single transformation needs THREE pieces: type (enlargement), scale factor (with sign), and centre of enlargement (as coordinates). Missing any one of the three loses an A1 mark every time.
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