https://file.aiquickdraw.com/mathbot/file/17270788934661uo3r3q1.jpg

The problem involves two similar triangles, \(\triangle ABC\) and \(\triangle XYZ\). To solve for the missing values of \(b\) and \(x\), we can use the properties of similar triangles. The sides of similar triangles are proportional.

### Step 1: Set up the proportion
The corresponding sides are:
- \( AB \) corresponds to \( XY \)
- \( BC \) corresponds to \( YZ \)
- \( AC \) corresponds to \( XZ \)

Using these correspondences, we can set up the proportion between corresponding sides.

#### For \(b\):
\[
\frac{AC}{XZ} = \frac{AB}{XY}
\]
Substituting the known values:
\[
\frac{b}{8} = \frac{15}{10}
\]
Now solve for \(b\):
\[
b = \frac{15}{10} \times 8 = 12
\]

#### For \(x\):
\[
\frac{BC}{YZ} = \frac{AB}{XY}
\]
Substituting the known values:
\[
\frac{9}{x} = \frac{15}{10}
\]
Now solve for \(x\):
\[
x = \frac{10}{15} \times 9 = 6
\]

### Final Answers:
- The value of \(b\) is **12**.
- The value of \(x\) is **6**.

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

Here are some related questions:
1. How do you verify that two triangles are similar?
2. What are some real-world applications of similar triangles?
3. Can all triangles be similar? Why or why not?
4. How do the angles in similar triangles compare?
5. How can the Pythagorean theorem help with solving triangle problems?

**Tip:** Always check that the ratios of corresponding sides remain consistent when solving problems involving similar triangles.

Determine the missing lengths on the similar triangles. The value of b is? The value of x is?

Learn how to determine the missing lengths in similar triangles using the properties of proportional sides. This problem demonstrates how to find the values of sides 'b' and 'x' by applying the similarity theorem and setting up ratios between corresponding sides. Suitable for students in Grades 8-10.

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