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The problem provided shows two triangles, \( \triangle ABC \) and \( \triangle DEF \), and asks to find \( x \) and \( y \) given that the triangles are similar (i.e., \( \triangle ABC \sim \triangle DEF \)).

### Step-by-step Solution:

1. **Understanding Triangle Similarity:**
   Since \( \triangle ABC \sim \triangle DEF \), the corresponding sides of the triangles are proportional. This means:
   \[
   \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}
   \]

2. **Identifying Given Side Lengths:**
   From the image, we have the following side lengths:
   - In \( \triangle ABC \):
     \[
     AB = 7, \quad BC = 6, \quad AC = 8
     \]
   - In \( \triangle DEF \):
     \[
     DE = 15, \quad DF = y, \quad EF = x
     \]

3. **Setting up Proportions:**
   Using the proportionality of similar triangles, we can set up the following equations:

   For side \( AB \) and \( DE \):
   \[
   \frac{AB}{DE} = \frac{7}{15}
   \]
   
   For side \( BC \) and \( EF \):
   \[
   \frac{BC}{EF} = \frac{6}{x}
   \]
   Thus, \( \frac{6}{x} = \frac{7}{15} \).

   For side \( AC \) and \( DF \):
   \[
   \frac{AC}{DF} = \frac{8}{y}
   \]
   Thus, \( \frac{8}{y} = \frac{7}{15} \).

4. **Solving for \( x \) and \( y \):**
   - For \( x \), solving \( \frac{6}{x} = \frac{7}{15} \):
     \[
     x = \frac{6 \times 15}{7} = \frac{90}{7} \approx 12.86
     \]
   
   - For \( y \), solving \( \frac{8}{y} = \frac{7}{15} \):
     \[
     y = \frac{8 \times 15}{7} = \frac{120}{7} \approx 17.14
     \]

### Final Answer:
- \( x \approx 12.86 \)
- \( y \approx 17.14 \)

Let me know if you'd like more details or further clarifications! 

Here are 5 related questions you might find helpful:
1. What are the properties of similar triangles?
2. How do we use the ratios of sides to determine missing side lengths?
3. Can angles also help determine similarity between triangles?
4. How does scaling of side lengths relate to area in similar triangles?
5. How can you apply this method to non-right triangles?

**Tip:** When dealing with similar triangles, always match corresponding sides and keep ratios consistent.

Given that ∆ABC ∼ ∆DEF, find x and y.

Learn how to find the values of x and y by using the properties of similar triangles. This solution covers setting up proportions and applying the triangle similarity theorem. Suitable for students in Grades 9-11.

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