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The problem shows that triangle \( A'B'C' \) is a dilation of triangle \( ABC \), and we are asked to find the length of \( AB \).

### Information given:
- The side \( A'C' \) in the smaller triangle is 10, and the corresponding side \( AC \) in the larger triangle is 15.
- The side \( A'B' \) in the smaller triangle is 6, and we are asked to find \( AB \) in the larger triangle.

### Step-by-Step Approach:
Dilation means that the sides of the triangles are proportional. The ratio between corresponding sides of the triangles remains constant. So, we can set up a proportion:

\[
\frac{A'B'}{AB} = \frac{A'C'}{AC}
\]

Substitute the known values:

\[
\frac{6}{AB} = \frac{10}{15}
\]

Simplify the right-hand side:

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

Now, solve for \( AB \):

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

Cross-multiply:

\[
6 \times 3 = 2 \times AB
\]

\[
18 = 2 \times AB
\]

\[
AB = \frac{18}{2} = 9
\]

### Answer:
The length of \( AB \) is \( 9 \), so the correct option is \( \text{C. } 9 \).

Would you like further details or have any questions?

### Related Questions:
1. How do you determine the scale factor in a dilation?
2. What are the properties of dilated triangles?
3. How can you verify that two triangles are similar using proportions?
4. What would happen to the angle measures in a dilation?
5. How do you apply dilation in coordinate geometry?

### Tip:
When working with dilation problems, always check for the ratio between corresponding sides. This is key to setting up the correct proportion!

Triangle A'B'C' is a dilation of triangle ABC. What is AB?

Solve the dilation problem where triangle A'B'C' is a dilation of triangle ABC. Learn how to use proportions and dilation theorem to find the unknown side AB. Suitable for students in Grades 8-10.

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