Lesson 7: Similar Figures

Property

We can also define similarity using transformations. Two figures are similar if you can map one exactly onto the other using a sequence that includes a dilation (to match the size) followed by any rigid motions (translation, reflection, or rotation to match the position).

• Congruent = Rigid Motions only.
• Similar = Dilation + Rigid Motions.

Examples

• Mapping $\Delta ABC$ to $\Delta A'B'C'$:
  • Step 1 (Size): First, look at the sizes. If $A'B'C'$ is twice as big, dilate $\Delta ABC$ by a scale factor of $k = 2$.
  • Step 2 (Position): Now that they are the same size, how do we get the intermediate triangle to park in the final spot? Translate it $3$ units Right and $1$ unit Down.
  • The sequence is: Dilation ($k=2$), then Translation.

Explanation

This skill is like solving a two-step puzzle. Always fix the SIZE first! Calculate your scale factor $k$ by comparing one pair of sides. Once you perform the mental dilation, your shape is now a congruent "clone" of the target. Your second step is simply deciding whether you need to slide it, flip it, or turn it to make it snap perfectly into place.

Property

Two figures are similar if, and only if:

1. Their corresponding angles are equal, and
2. Their corresponding sides are proportional.

Examples

• A rectangle with sides 4 cm and 6 cm is similar to a rectangle with sides 8 cm and 12 cm. The scale factor is 2.

• A photograph measuring 4 inches by 6 inches is enlarged to 8 inches by 12 inches. The enlarged photo is similar to the original.