Skill: Use transformations to establish congruence and similarity

A transformation shows how one figure can be moved to match another.

• Translations, rotations, and reflections preserve side lengths and angle measures. If these transformations map one figure exactly onto another, the figures are congruent.
• A dilation changes lengths by the same scale factor but preserves angle measures. If rigid motions and one dilation map one figure onto another, the figures are similar.

Example: Show that $\triangle ABC$ is similar to $\triangle A'B'C'$.

The coordinates are

$$A(0,0),\quad B(2,0),\quad C(0,1)$$

and

$$A'(3,2),\quad B'(7,2),\quad C'(3,4).$$

Step 1: Apply a dilation.

Dilate $\triangle ABC$ by a scale factor of $2$ centered at the origin. The coordinate rule is

$$(x,y)\longrightarrow(2x,2y).$$

So the points become

$$\begin{aligned} A(0,0)&\longrightarrow A_1(0,0),\\ B(2,0)&\longrightarrow B_1(4,0),\\ C(0,1)&\longrightarrow C_1(0,2). \end{aligned}$$

Step 2: Apply a translation.

Translate every point $3$ units right and $2$ units up. The rule is

$$(x,y)\longrightarrow(x+3,y+2).$$

Therefore,

$$\begin{aligned} A_1(0,0)&\longrightarrow A'(3,2),\\ B_1(4,0)&\longrightarrow B'(7,2),\\ C_1(0,2)&\longrightarrow C'(3,4). \end{aligned}$$

The transformed triangle matches $\triangle A'B'C'$, so

$$\triangle ABC\sim\triangle A'B'C'.$$

The dilation multiplies every side length by $2$, and the translation does not change lengths or angles. Thus, corresponding angles are equal, and corresponding side lengths have the constant scale factor $2$.

For congruence, the same process uses only translations, rotations, and reflections, with no change in scale.