We are told that △ABC ~ △DFE. Because the triangles are similar, their corresponding sides are proportional.

Step 1: Set up the proportion.

From the image:

• $AB = 18$, $BC = 15$, $AC = 21$ in △ABC.
• $DF = 7$, $FE = x$ (unknown), $DE$ is not required.

The corresponding sides of the triangles are proportional, so we can write:

$\frac{BC}{FE} = \frac{AB}{DF}$

Substitute the given values:

$\frac{15}{x} = \frac{18}{7}$

Step 2: Solve for $x$.

To isolate $x$, cross-multiply:

$15 \cdot 7 = 18 \cdot x$

Simplify:

$105 = 18x$

Divide both sides by 18:

$x = \frac{105}{18} \approx 5.83$

Final Answer:

$x \approx 5.83$

Do you want further clarification on this problem, or have other questions? Let me know!

Here are 5 related questions you might find useful:

1. How do you determine if two triangles are similar?
2. What are the properties of similar triangles?
3. How do you solve proportions in triangles involving unknown sides?
4. What are other methods to prove triangle similarity (e.g., AA, SAS, SSS)?
5. How do you apply ratios in real-world problems involving geometry?

Tip: Always match corresponding sides carefully when setting up proportions in similar triangles.