G19, R12 - Congruence and similar shapes

Congruent or similar

A rectangular photograph measures $12\,\mathrm{cm}$ by $8\,\mathrm{cm}$. Can it fill an $18\,\mathrm{cm}$ by $12\,\mathrm{cm}$ frame without stretching or cropping? Both dimensions must grow by the same factor. Making just the width fit could leave you with a rather wide-looking face.

Think of putting one cut-out exactly over another. You may move it, turn it or flip it. If it fits exactly, the shapes are congruent. If you must also change its size, they may be similar.

The word corresponding means matching positions in the two shapes. A short side matches the corresponding short side, even if the second shape is turned round. Orientation does not decide congruence or similarity.

[DIAGRAM: asset_name: Matching rectangles and a distorted rectangle; asset_slug: aqa8300_g17_shape_comparison; caption: The first two rectangles have matching side ratios, while the third has been stretched in only one direction; description: Three separated rectangles, with accurate proportions, labelled A 12 cm by 8 cm, B 18 cm by 12 cm, and C 18 cm by 10 cm. A is blue, B is sage and C is clay. Below A and B the matching ratios 18 over 12 equals 12 over 8 equals 1.5 are shown. Below C a note says the length factors do not match.]

Equal angles alone do not prove that all polygons are similar. Every rectangle has four right angles, yet a long thin rectangle is not similar to a square. For triangles, equal corresponding angles do suffice: two matching angles force the third, giving the same shape. This is the AA similarity test.

Equal areas do not prove similarity either. Rectangles measuring $6\,\mathrm{cm}$ by $4\,\mathrm{cm}$ and $8\,\mathrm{cm}$ by $3\,\mathrm{cm}$ both have area $24\,\mathrm{cm}^2$, but their long-side-to-short-side ratios are $3:2$ and $8:3$. The same amount of card can make quite different shapes.

Use both the shape and the size information when judging the next claim.

Corresponding lengths

Write the matching side pairs before calculating a scale factor, especially when a diagram has been turned or reflected.

If $ABC$ is similar to $PQR$, the order tells you that $A$ matches $P$, $B$ matches $Q$ and $C$ matches $R$. Therefore $AB$ matches $PQ$, $BC$ matches $QR$ and $AC$ matches $PR$. Sketch the shapes separately if that helps.

Start with a pair of lengths you know, then use that one factor for the missing pair.

Worked example 2 changes the direction: the unknown length belongs to the smaller shape. The factor can be less than $1$; it is still an enlargement scale factor in the mathematical sense.

The common error is to transfer an increase rather than a factor.

For the opening photograph, both factors are $18\div12=1.5$ and $12\div8=1.5$. It fills the frame exactly, with no distortion. Now try a different reduction.

Similar triangles inside a figure

A line parallel to one side of a triangle makes a smaller triangle of the same shape at the opposite vertex.