Congruence & similarity

Zoom a photo larger and it stays the same picture, just bigger. Lay two identical parts on top of each other and they match exactly. These two ideas are what congruence and similarity are about, and they are far more useful than they sound: with them you can even measure the height of a tree without climbing it.

What does congruent mean?

Two figures are congruent if they cover each other exactly: same shape and same size. You can slide, turn or flip one so it lands precisely on the other. Corresponding sides have the same length, corresponding angles the same size. A triangle stays congruent to itself no matter how you turn it or flip it over on the paper.

Similar: same shape, usually a different size

Similar figures have the same shape, usually in a different size. The angles stay equal, all sides are stretched or shrunk by the same factor. A photo and its enlargement are similar, a model car and the real car too. Congruent figures are a special case of this: similar with scale factor 1, so with no change in size at all.

The scale factor

The scale factor $k$ tells you how strongly a figure is enlarged. If $k = 3$, every side becomes three times as long. You find it by dividing a new length by the matching old length. If $k$ is greater than 1 the figure grows, if $k$ is smaller than 1 it shrinks.

The intercept theorem

When two rays starting at the same point cut two parallel lines, two similar triangles appear. Corresponding segments then share the same ratio: $\frac{a}{b} = \frac{a'}{b'}$. That is exactly what measuring with shadows relies on. Sunlight arrives practically parallel, so pole plus shadow and tree plus shadow form two similar triangles. If a 1 m pole casts a 2 m shadow and the tree beside it casts 12 m at that same moment, the tree is 6 m tall, because height to shadow stays 1 to 2.