https://file.aiquickdraw.com/mathbot/file/1727370152365q0hk3x9g.jpg

In the image, we are dealing with two similar triangles, where the smaller triangle is a dilation of the larger triangle centered at point O. The side lengths of the larger triangle are given as 10, 12, and 15, while the smaller triangle has one side marked as 6, and two unknowns, \( x \) and \( y \).

We need to:

1. Find the scale factor of the dilation.
2. Find the lengths of \( x \) and \( y \).

### Step 1: Find the Scale Factor

The corresponding sides of similar triangles are proportional. So, we can find the scale factor by comparing the corresponding sides of the two triangles. Let's compare the side of length 6 in the smaller triangle with the corresponding side of length 15 in the larger triangle.

\[
\text{Scale Factor} = \frac{6}{15} = \frac{2}{5}
\]

Thus, the scale factor is \( \frac{2}{5} \).

### Step 2: Find the Lengths of \( x \) and \( y \)

Since the triangles are similar, the ratios of the corresponding sides are equal. Using the scale factor \( \frac{2}{5} \), we can find \( x \) and \( y \) by multiplying the corresponding sides of the larger triangle by the scale factor.

- \( x \) corresponds to the side of length 12 in the larger triangle:
  \[
  x = \frac{2}{5} \times 12 = \frac{24}{5} = 4.8
  \]

- \( y \) corresponds to the side of length 10 in the larger triangle:
  \[
  y = \frac{2}{5} \times 10 = 4
  \]

### Final Answer:
- **Scale Factor**: \( \frac{2}{5} \)
- **Length of \( x \)**: \( 4.8 \)
- **Length of \( y \)**: \( 4 \)

Would you like further details or have any questions about the solution?

Here are five related questions for further exploration:
1. How do we determine corresponding sides in similar triangles?
2. Can we use this method for triangles with angles instead of side lengths?
3. What are some real-life applications of dilation in geometry?
4. How do we calculate the area ratio between similar triangles?
5. What happens to the perimeter ratio of two similar triangles?

**Tip**: Always double-check that the sides you are comparing in similar triangles are indeed corresponding ones.

In the following figure, the smaller triangle is the image of the larger triangle under a dilation centered at point O. Find the scale factor and the length of x and y as pictured.

Learn how to find the scale factor and side lengths of similar triangles under a dilation centered at a point. This problem involves key geometry concepts like dilation, similar triangles, and proportions. Suitable for students in Grades 8-10.

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