Skill: Apply proportional reasoning to similar triangles

Similar triangles have the same shape, so their corresponding angles are equal and their corresponding sides are in the same ratio. The triangles may be turned or flipped, so match sides by their corresponding angles—not by their position on the page.

Suppose $\triangle ABC \sim \triangle DEF$, with the correspondence

$$A \leftrightarrow D,\qquad B \leftrightarrow E,\qquad C \leftrightarrow F.$$

This means the matching sides are

$$AB \leftrightarrow DE,\qquad BC \leftrightarrow EF,\qquad AC \leftrightarrow DF.$$

Example

In two similar triangles:

• $AB=6$ cm
• $BC=9$ cm
• $DE=10$ cm

Find $EF$.

Step 1: Identify corresponding sides.

Since $AB \leftrightarrow DE$ and $BC \leftrightarrow EF$, use those pairs. Do not compare $AB$ with $EF$, because they are not corresponding sides.

Step 2: Find the scale factor.

The side $AB$ grows from $6$ cm to $10$ cm:

$$\text{scale factor}=\frac{DE}{AB}=\frac{10}{6}=\frac{5}{3}.$$

Step 3: Apply the same scale factor to the matching side.

The side $BC$ corresponds to $EF$:

$$EF=BC\left(\frac{5}{3}\right) =9\left(\frac{5}{3}\right) =15.$$

Therefore,

$$\boxed{EF=15\text{ cm}}.$$

You can also set up a proportion directly:

$$\frac{DE}{AB}=\frac{EF}{BC} \quad\Rightarrow\quad \frac{10}{6}=\frac{EF}{9}.$$

Both methods give the same answer. Always make sure each ratio compares corresponding sides in the same order.