Similarity Proofs & Indirect Measurement

Naming the proportion from the similarity statement

The order of the letters carries the information. If $\triangle ABC \sim \triangle XYZ$, then $A$ matches $X$, $B$ matches $Y$, and $C$ matches $Z$. Corresponding sides are the ones between matching vertices: $AB$ pairs with $XY$, $BC$ with $YZ$, and $AC$ with $XZ$.

So the proportion you can always write is $\dfrac{AB}{XY} = \dfrac{BC}{YZ} = \dfrac{AC}{XZ}$. Read the statement carefully before you set up the fractions — swapping two letters gives a proportion that looks reasonable but is wrong.

How indirect measurement works

A person and their shadow form a small triangle; a flagpole and its shadow form a large one. The sun's rays hit both at the same angle, and both stand vertical, so the triangles are similar by AA.

That means $\dfrac{\text{person height}}{\text{person shadow}} = \dfrac{\text{pole height}}{\text{pole shadow}}$. Three of those four lengths are easy to measure, so you solve the proportion for the fourth — the height you could not reach.

The figure shows the two similar right triangles: the sun strikes both at the same angle (the marked angle) and both objects stand vertical, so they are similar by AA.

Worked examples

Try one yourself

Common questions