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To solve for \( x \) and \( y \), we can use the property of similar triangles. The given diagram shows two similar triangles. For similar triangles, the corresponding sides are proportional.

The two similar triangles are:

1. The smaller triangle with sides \( x \) (left side), \( 8 \) cm (right side), and \( 10 \) cm (base).
2. The larger triangle with sides \( 6 \) cm (left side), \( y \) (right side), and \( 15 \) cm (base).

Using the property of similar triangles, we have the following proportion relationships:

\[
\frac{x}{6} = \frac{8}{y} = \frac{10}{15}
\]

Let's first find the common ratio using the base lengths:

\[
\frac{10}{15} = \frac{2}{3}
\]

Now we can use this ratio to find \( x \) and \( y \).

1. **Finding \( x \):**

\[
\frac{x}{6} = \frac{2}{3}
\]

Solving for \( x \):

\[
x = 6 \times \frac{2}{3} = 4 \text{ cm}
\]

2. **Finding \( y \):**

\[
\frac{8}{y} = \frac{2}{3}
\]

Solving for \( y \):

\[
y = \frac{8 \times 3}{2} = 12 \text{ cm}
\]

Hence, the values are:
- \( x = 4 \) cm
- \( y = 12 \) cm

Would you like further details or have any questions?

### 5 Related Questions:
1. How do we prove two triangles are similar?
2. What is the general formula for the ratio of sides in similar triangles?
3. How can we apply this concept to solve real-world problems?
4. How do we handle cases when the triangles are not similar?
5. What are other geometric properties related to similar triangles?

### Tip:
For any pair of similar figures, always ensure you identify the corresponding sides before setting up proportions.

Find x and y in the given diagram of similar triangles.

Learn how to find the side lengths of similar triangles using proportions. This detailed guide includes problem-solving steps, formulas, and explanations to enhance understanding of geometry concepts suitable for students in Grades 8-10.

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