Similar figures and scale factors, the AA, SSS and SAS similarity criteria, the triangle proportionality theorem, perimeter and area ratios, and indirect measurement.
Same shape, different size
A photo and its enlargement, a map and the land it shows, a model car and the real one: same shape, different size. In geometry, such figures are similar, written with the symbol \(\sim\).
As with congruence, the order of letters matters: \(\triangle ABC \sim \triangle DEF\) means \(A\) matches \(D\), and so on.
Proving triangles similar
AA is by far the most used: two matching angles force the third to match too.
The triangle proportionality theorem
The parallel line also cuts off a small triangle \(\triangle ADE\) that is similar to \(\triangle ABC\), by AA (corresponding angles).
Perimeters and areas
Indirect measurement
Sunlight arrives in parallel rays, so at the same moment an object and its shadow form a right triangle similar to the one formed by any other upright object and its shadow.
Practice
Quick check
Which information is enough to prove two triangles similar?
That's the AA criterion.
Quick check
Two similar triangles have scale factor \(3\). What is the ratio of their perimeters?
Perimeters scale like lengths, by \(k\). Only areas scale by \(k^2\).
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