What you’ll learn
Determine similarity using transformations.
Key idea
Rigid motions preserve lengths; a positive dilation scales all corresponding lengths uniformly and preserves similarity.
Nonunit dilation preserves similarity but changes size and therefore is not congruence.
Worked example
Enter the exact value or ordered values in the format requested below.
A rotation followed by dilation factor 2 maps side AB=4 cm. Enter corresponding image side length with unit.
- The rigid rotation keeps original side length 4 cm unchanged.
- Dilation factor 2 gives image length 4×2=8 cm; undoing the dilation gives 8/2=4 cm.
- Because the positive factor differs from one, shape is preserved but corresponding size changes: similar, not congruent.
Answer: 8 cm
Quick Check
Evaluate a worked attempt at determine similarity using transformations, identify the first invalid step, and provide the correction.
Side AB=8 cm undergoes rotation then dilation factor 3. Step 1 says rotation preserves 8 cm. Step 2 calls the final image congruent, ignoring dilation. Locate the first invalid step and correct the relationship.