Similar Figures

In mathematics, we say that two objects are similar if they have the same shape, but not necessarily the same size. This means that we can obtain one figure from the other through a process of expansion or contraction, possibly followed by translation, rotation or reflection. If the objects also have the same size, then they are congurent.

For a general shape, it can be tricky to show that two items are similar. We have to check all corresponding angles and ratios of side lengths before we can reach a conclusion. For example, a $1 \times 2$ rectangle is not similar to a $2 \times 3$ rectangle, even though they both have 4 right angles, since their side lengths have different ratios.

In the case of the triangle, we have slightly more information due to the Sine rule and Cosine Rule. We also have the methods SSS (side-side-side), SAS (side-angle-side), ASA (angle-side-angle), AAS (angle-angle-side) and AAA (angle-angle-angle), to prove two triangles are similar. Note that in comparison with congruent figures, side here refers to having the same ratio of side lengths. Also, note that method AAA is equivalent to AA, since the sum of angles in a triangle is equal to $180^\circ$.

The following result is immediate by expanding the formulas for perimeter and ratio:

If two figures $F_1$ and $F_2$ are similar with a corresponding side length ratio of $R$, then the ratio of their perimeters is $R$, and the ratio of their areas is $R^2$.

For a useful case of similar triangles, see Parallel Lines Property C.

Worked Examples

1. Show that all circles are similar to each other.

If we lay the circles over each other such that their centers coincide, then the expansion by factor $\frac{R_2} {R_1}$ (where $R_1$ and $R_2$ are the radii of the two circles) maps the first circle to the second.

Note: In a similar fashion, we can show that all regular polygons with the same number of edges are similar to each other. Why doesn't this work for rectangles?

2. Prove that if 2 triangles have equal corresponding angles, then they are similar.

Let triangles $ABC$ and $DEF$ have equal corresponding angles.

Translate $DEF$ such that $A= D$.

Since $\angle BAC = \angle EDF$, rotate $DEF$ about $A$ such that $E$ lies on $AB$ and $F$ lies on $AC$.

Since $\angle ABC = \angle DEF$, it follows that $BC$ and $EF$ are parallel, and hence we have a scaling map which brings $ABC$ to $DEF$. Thus these two triangles are similar.

3. Similar triangles $T_1$ and $T_2$ have perimeters of $100$ and $200$ respectively. If the area of $T_1$ is 200, what is the area of $T_2$?

Using the above result, the ratio of expansion between these triangles is $R = \frac{200}{100} = 2$. Hence, the ratio of their areas is $R^2 = 2^2 = 4$, and thus the area of $T_2$ is $4 \times 200 = 800$.