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.