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The problem involves finding the scale factor used in a dilation between two triangles: a smaller triangle \( \triangle ABC \) (pre-image) and a larger triangle \( \triangle A'B'C' \) (image).

From the diagram, the sides of the triangles are given as:
- In \( \triangle ABC \), the legs are 4 and 3.
- In \( \triangle A'B'C' \), the corresponding legs are 16 and 12.

The scale factor \( k \) in a dilation can be found by dividing the corresponding sides of the image triangle by the pre-image triangle.

For the side corresponding to \( AB \) and \( A'B' \):
\[
k = \frac{A'B'}{AB} = \frac{16}{4} = 4
\]

Similarly, for the side corresponding to \( BC \) and \( B'C' \):
\[
k = \frac{B'C'}{BC} = \frac{12}{3} = 4
\]

Since the ratio is consistent for both pairs of corresponding sides, the scale factor used in the dilation is **4**.

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:
1. How do you calculate the scale factor if the center of dilation is not the origin?
2. What happens to the area of a shape when a dilation with a given scale factor is applied?
3. Can a dilation result in a reduction of size? How does the scale factor change in such a case?
4. What are the properties of corresponding angles in dilations?
5. How can you prove that the triangles in a dilation are similar?

**Tip:** To verify dilations, you can always check that the ratios of corresponding sides are equal, confirming the uniform scaling of the figure.

The picture shows an image and its pre-image. What scale factor was used in the dilation shown?

Learn how to find the scale factor used in a dilation between two triangles. In this example, a smaller triangle with sides 3 and 4 is dilated to a larger triangle with sides 12 and 16. The scale factor is calculated by dividing corresponding sides of the triangles, resulting in a factor of 4.

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