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!