Similar Polygons & Corresponding Parts

Matching up corresponding parts

When the similarity is written with letters, the order of the letters is the matching. If triangle $ABC \sim$ triangle $DEF$, then $A$ pairs with $D$, $B$ with $E$, and $C$ with $F$ — so side $AB$ corresponds to side $DE$, and side $BC$ corresponds to side $EF$. In the figure, the matching arc marks show corresponding angles are equal: one arc at $A$ and $D$, two at $B$ and $E$, three at $C$ and $F$.

When you only have a picture, match by position: shortest side with shortest side, longest with longest, and the sides touching equal angles with each other. A right angle in each figure is a helpful anchor — the sides that form it correspond to the sides that form the other one.

Writing the proportion

Put matching sides in matching places. One clean setup: each fraction compares the big figure to the small figure, so both numerators come from one polygon and both denominators come from the other. For triangles with heights $10$ and $4$ and bases $x$ and $6$, that gives $\dfrac{10}{4} = \dfrac{x}{6}$.

Then cross multiply and solve. Any consistent setup works — you can also make each fraction a ratio within one figure, like $\dfrac{x}{10} = \dfrac{6}{4}$ — as long as you never mix the pattern halfway through.

The scale factor shortcut

If one pair of corresponding sides gives you a clean whole-number ratio, skip the proportion. Compute the scale factor $k$ from that pair, then multiply the known side by $k$. If $AB = 6$ and its corresponding side $DE = 18$, then $k = 3$, and every other side of the big triangle is triple its match in the small one.

Both methods are the same math. The proportion is safer when the ratio is messy; the scale factor is faster when it's clean.

Worked examples

Try one yourself

Common questions