To find $DE$ given that $\triangle ABC \sim \triangle DFE$, we use the properties of similar triangles. Similar triangles have corresponding sides in proportion.

Let’s denote the sides of the triangles as follows:

• $AB$ and $DE$ are corresponding sides.
• $BC$ and $EF$ are corresponding sides.
• $CA$ and $FD$ are corresponding sides.

The problem involves finding the length of $DE$. Let's assume we know the lengths of some sides of the triangles. For example:

$AB = a, \quad BC = b, \quad CA = c$ $DE = x, \quad EF = y, \quad FD = z$

Given that the triangles are similar, the ratio of the lengths of corresponding sides is the same. Thus, we have:

$\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}$

Let's say we are given specific lengths for some of these sides. Suppose:

• $AB = 6$
• $BC = 8$
• $CA = 10$
• $DE = x$
• $EF = 12$
• $FD = 15$

Using the proportionality of the sides, we get:

$\frac{AB}{DE} = \frac{BC}{EF}$

Substituting the known values:

$\frac{6}{x} = \frac{8}{12}$

Solving for $x$:

$\frac{6}{x} = \frac{2}{3}$

Cross-multiplying:

$2x = 18$

$x = 9$

Therefore, $DE = 9$.

If we need to use a different ratio, we can confirm using another pair of corresponding sides. Let's check using $\frac{CA}{FD}$:

$\frac{10}{15} = \frac{2}{3}$

Since both calculations give consistent results, the length of $DE$ is indeed $9$.

Do you want more details or have any questions?

Here are 8 related questions for further exploration:

1. What is the definition of similar triangles?
2. How do you determine if two triangles are similar?
3. What are the properties of similar triangles?
4. How do you find the scale factor between two similar triangles?
5. What is the relationship between the areas of similar triangles?
6. How do corresponding angles relate in similar triangles?
7. Can the Pythagorean Theorem be applied to similar triangles?
8. How can similar triangles be used in real-world applications?

Tip: When solving problems involving similar triangles, always set up a proportion based on the corresponding sides and solve for the unknown length.