Use transformations to establish congruence and similarity

Congruence is established by showing that one plane figure can be mapped onto another through translations, rotations, and reflections, which preserve distances and angle measures. Similarity is established through a sequence of these rigid motions together with a dilation, so corresponding angles are equal and corresponding lengths share one constant scale factor; coordinate rules and diagrams can represent these mappings. The scope excludes three-dimensional transformations, abstract transformation groups, and matrix-based generalizations.

Detailed Explanation: 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.

Learn by doing: Use transformations to establish congruence and similarity

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Multiple Transformations - Grid Image & Transformations to Similar or Congruent

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Use transformations to establish congruence and similarity

Detailed Explanation: Use transformations to establish congruence and similarity

Learn by doing: Use transformations to establish congruence and similarity

Practice with unlimited practice problems

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Multiple Transformations - Grid Image & Transformations to Similar or Congruent