Similarity

What is a Similarity?

Examine the rectangles shown below. Although their sizes differ, they have the same shape because the proportions of their side lengths are identical.

When $\textcolor{colordef}{A}$ is enlarged to form $\textcolor{colorprop}{A'}$, the side lengths are doubled. The scale factor is $\textcolor{olive}{2}$.

Definition Similarity and Enlargement/Reduction

A similarity with scale factor $k>0$ is a transformation that multiplies all distances by the same number $k$.

If $k>1$, the similarity is an enlargement.
If $0 < k < 1$, the similarity is a reduction.
If $k=1$, the similarity preserves all distances (it is an isometry), so the figure has the same size and shape.

Theorem Fundamental Transformations Similarity Theorem

Any similarity can be expressed as the composition of one or more fundamental transformations (reflection, translation, rotation, and homothety).

Example

The blue $\textcolor{colorprop}{L}$ is similar to the red $\textcolor{colordef}{L}$: it is a reduction of it.

The blue $\textcolor{colorprop}{L}$ is the image of the red $\textcolor{colordef}{L}$ through a homothety of scale factor $0.5$ ($\textcolor{colordef}{L} \to \textcolor{olive}{L'}$) followed by a rotation ($\textcolor{olive}{L'} \to \textcolor{colorprop}{L}$).

Similar Figures

Definition Similar Figures

Two figures are similar if one can be obtained from the other by a similarity (an enlargement, a reduction, or an isometry).

Discover

The figure $F'$ is an enlargement of the figure $F$ by a scale factor of $2$.

The ratios of the corresponding sides are:

$\dfrac{\textcolor{colorprop}{A'B'}}{\textcolor{colordef}{AB}} = \dfrac{2 \times 4~\text{cm}}{4~\text{cm}} = \textcolor{olive}{2}$
$\dfrac{\textcolor{colorprop}{A'C'}}{\textcolor{colordef}{AC}} = \dfrac{2 \times 3~\text{cm}}{3~\text{cm}} = \textcolor{olive}{2}$
$\dfrac{\textcolor{colorprop}{B'C'}}{\textcolor{colordef}{BC}} = \dfrac{2 \times 5~\text{cm}}{5~\text{cm}} = \textcolor{olive}{2}$

Thus, the ratios of the corresponding sides are equal to the scale factor.

Proposition Properties of Similar Figures

For similar figures:

The ratios of the lengths of corresponding sides are all equal to the same scale factor.
The corresponding angles are equal.

Example

The figures $\textcolor{colordef}{F}$ and $\textcolor{colorprop}{F'}$ are similar. Find $x$.

Answer

The ratios of the corresponding sides are equal:$$\begin{aligned}\dfrac{\textcolor{colorprop}{A'C'}}{\textcolor{colordef}{AC}} &= \dfrac{\textcolor{colorprop}{A'B'}}{\textcolor{colordef}{AB}} \\ \dfrac{x}{3} &= \dfrac{5}{4} \\ x &= 3 \times \dfrac{5}{4} \\ x &= 3.75~\text{cm}\end{aligned}$$