Similar Triangles

≈ 25 min

Two triangles are similar (written $\triangle ABC \;|||\; \triangle DEF$ ) when they have the same shape but not necessarily the same size — one is a scaled copy of the other. For similar triangles:

corresponding angles are equal, and
corresponding sides are in the same ratio (the scale factor).

Worked example. A triangle has sides $3, 4, 5$ . A similar triangle has its shortest side $9$ . The scale factor is $\dfrac{9}{3} = 3$ , so every side is 3 times larger: the other sides are $4 \times 3 = 12$ and $5 \times 3 = 15$ .

Similarity is what lets us use a small triangle to measure a huge one we could never reach.

Similar triangles always have…

equal angles and proportional sides

equal sides only

equal area

equal perimeter

Triangle A has sides $3, 4, 5$ . A similar triangle B has its shortest side equal to $9$ . What is the scale factor from A to B?

Using the scale factor of 3 from the previous question, find the longest side of triangle B (triangle A's longest side is 5).

A person 2 m tall casts a 3 m shadow. At the same time a flagpole casts a 12 m shadow. How tall (in metres) is the flagpole?