Triangle Similarity: Ratios and Applications

Two geometric figures are similar when they have the same shape, but not necessarily the same size. Think of a photo and its enlarged version: same image, different sizes. For triangles, similarity means equal corresponding angles and proportional corresponding sides.

The similarity ratio k is the constant proportion between corresponding sides. If triangle ABC has sides 3, 4, 5 and triangle DEF has sides 6, 8, 10, then k = 2 (DEF is twice ABC). Every side of DEF is exactly twice the length of the corresponding side of ABC.

It is important not to confuse similarity with congruence. Congruent figures have the same shape AND the same size (k = 1). Similar figures have the same shape but possibly different sizes (k ≠ 1). Every congruence is a similarity, but not every similarity is a congruence.

AA criterion (Angle-Angle): if two angles of one triangle equal two angles of another, the triangles are similar. Because the interior angles of a triangle always sum to 180°, two equal angles guarantee that the third is also equal.

SAS criterion (Side-Angle-Side): if two sides of one triangle are proportional to two sides of another, and the included angle is equal, the triangles are similar. The angle must be the one formed by exactly the two proportional sides.

SSS criterion (Side-Side-Side): if all three sides of one triangle are proportional to the three sides of another (the same similarity ratio k for all pairs), the triangles are similar. This criterion requires checking all three pairs of sides.

If the similarity ratio between two triangles is k, then the ratio of their perimeters is also k, but the ratio of their areas is k². This is fundamental: if a triangle has sides twice as long (k = 2), it has a perimeter twice as large but an area four times as large.

To find unknown sides, set up a proportion using the known corresponding sides. If ABC ~ DEF with k = 3, and side AB = 5 cm, then DE = 5 × 3 = 15 cm. If DE = 12 and AB = 4, then k = 3; if side BC = 7, then EF = 21.

A classic use is calculating the height of objects using shadows. If a 1 m stake casts a 1.5 m shadow at the same moment that a building casts a 30 m shadow, the triangles formed by the stake and the building with their sun rays are similar. Therefore: building height / 1 m = 30 m / 1.5 m, so height = 20 m.

In cartography, maps are similar models of the earth's surface. A scale of 1:100,000 means that 1 cm on the map represents 100,000 cm = 1 km in reality. The similarity ratio is k = 1/100,000. To find real distances, measure on the map and multiply by the inverse scale.

In photography and cinema, lenses create projections that obey triangle similarity. The relationship between the size of the object, its distance from the camera and the size of the image formed is given directly by the similarity ratio. Photographers and art directors use this principle to calculate perspectives and camera angles.