5.2 Similar Polygons
Two polygons are similar if their corresponding angles are congruent and if their corresponding sides are proportional. When referring to similar polygons, we say that there is a ratio of proportionality or a scale factor between the two polygons. For example, if two right triangles are similar with a scale factor of $2:3$, then all corresponding angles are congruent but all corresponding sides are in the ratio of $2:3$. If the hypotenuse of the smaller triangle has a length of 4 then the length of the hypotenuse of the larger triangle is 6. The ratio $\frac{4}{6}$ is equivalent to $\frac{2}{3}$.
It is important to remember that if two triangles are in a ratio of $2:3$, this does not mean that the corresponding sides need to be 2 and 3. If we know the ratio of similitude between two triangles is $x:y$, then corresponding sides of the two triangles satisfy the proportion $$\frac{ax}{ay}=\frac{bx}{by}=\frac{cx}{cy}.$$
When we say that two figures are similar — for example, two triangles — we have a special notation ($\sim$). We will say $\triangle ABC \sim \triangle PQR$ and as we do with congruent triangles, will refer to corresponding sides and corresponding angles. Notice here that if $\triangle ABC \sim \triangle PQR$, then corresponding angles are congruent and corresponding sides satisfy the proportion $$\frac{ax}{ay}=\frac{bx}{by}=\frac{cx}{cy}$$ and each fraction reduces to $\frac{x}{y}$, which is then the ratio of proportionality or the scale factor.
Example:
If the rectangles $ABCD$ and $PQRS$ to the right are similar, what is
the scale factor? Find the value of $x$. Solution: The scale factor is the ratio between any pair of corresponding sides, so the scale factor is $8:12$, or$2:3$. Since corresponding sides are all in the same ratio,$\frac{x}{9}=\frac{2}{3}$ so $x=6$. |