The scale factor between two similar triangles is the ratio of corresponding sides. Since the longest sides of the triangles correspond to each other, we can find the scale factor by dividing the length of side DE by the length of side AB.

Given:

• $AB = 12$ cm (the longest side of triangle ABC),
• $DE = 8$ cm (the longest side of triangle DEF).

The scale factor from triangle ABC to triangle DEF is:

$\text{Scale factor} = \frac{DE}{AB} = \frac{8}{12} = \frac{2}{3}$

So, the scale factor that takes triangle ABC to triangle DEF is $\frac{2}{3}$.

Would you like any more details on this, or have any other questions?

Here are 5 related questions:

1. How would you find the scale factor if the triangles were oriented differently?
2. What happens to the areas of similar triangles with a scale factor of $\frac{2}{3}$?
3. If side BC of triangle ABC is 9 cm, what is the length of side EF in triangle DEF?
4. Can the scale factor ever be greater than 1? What would it mean?
5. How can you use the scale factor to find unknown angles in similar triangles?

Tip: In similar triangles, corresponding angles are equal, and corresponding sides are proportional to the scale factor.