Congruence and Similarity through Transformations

In geometry, transformations are operations that move or resize a figure. By applying specific sequences of transformations, we can determine whether two figures are congruent or similar.

Congruence: Rigid Transformations

A rigid transformation changes the position or orientation of a figure without changing its size or shape. There are three types of rigid transformations:

• Translation (sliding)
• Rotation (turning)
• Reflection (flipping)

If you can map Figure A exactly onto Figure B using only a sequence of rigid transformations, the two figures are congruent. For congruent figures:

• Corresponding sides are exactly equal in length.
• Corresponding angles are exactly equal in measure.

Similarity: Adding Dilations

A dilation is a transformation that changes the size of a figure but keeps its shape, based on a scale factor $k$.

If you can map Figure A onto Figure B using a sequence of rigid transformations plus a dilation, the two figures are similar. For similar figures:

• Corresponding angles are exactly equal in measure.
• Corresponding side lengths are proportional. The ratio of their side lengths is equal to the scale factor $k$.

How to Prove Congruence or Similarity

To show the relationship between two figures, you need to describe the step-by-step sequence that maps one onto the other.

Example 1: Proving Congruence Suppose Triangle $A$ is mapped to Triangle $B$ by translating it $3$ units to the right and then reflecting it across the $x$-axis.

• Conclusion: Because translations and reflections are both rigid transformations (preserving size and shape), Triangle $A$ and Triangle $B$ are congruent.

Example 2: Proving Similarity Suppose Rectangle $C$ is mapped to Rectangle $D$ by rotating it $90^\circ$ clockwise and then dilating it by a scale factor of $k = 2$.

• Conclusion: Because the sequence includes a dilation, the size has changed but the shape remains the same. Therefore, Rectangle $C$ and Rectangle $D$ are similar, but not congruent.