Lesson 2: Similarity

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

If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths are in the same ratio.

Examples

• $\triangle ABC$ is similar to $\triangle XYZ$. If side $AB=6$ and corresponding side $XY=2$, the ratio is $\frac{6}{2}=3$. If side $BC=9$, then corresponding side $YZ$ must be $\frac{9}{3}=3$.

• A man who is 6 feet tall casts a 4-foot shadow. A nearby tree casts a 20-foot shadow. Let $h$ be the tree's height. The ratio is $\frac{h}{6} = \frac{20}{4}$, so $4h = 120$, and $h=30$ feet.