To find the scale factor used in the dilation of the triangle, we compare the lengths of corresponding sides from the original triangle to the dilated one.

From the image:

• The side $GI = 10 \, \text{cm}$ in the original triangle corresponds to $G'I' = 25 \, \text{cm}$ in the dilated triangle.
• The side $HG = 6 \, \text{cm}$ corresponds to $H'G' = 15 \, \text{cm}$.

The scale factor $k$ is the ratio of any corresponding side in the dilated triangle to the original triangle. We can use the side $GI$ and $G'I'$ to calculate this.

$k = \frac{G'I'}{GI} = \frac{25}{10} = 2.5$

Thus, the scale factor is $2.5$.

Let me know if you'd like further details or have any questions!

Here are 5 related questions:

1. How does dilation affect the angles of a triangle?
2. Can a dilation reduce the size of a figure, and if so, what would the scale factor look like?
3. If one side of the original triangle is 8 cm, what would its corresponding side be in the dilated triangle using this scale factor?
4. How do you find the area of the dilated triangle if you know the area of the original triangle?
5. What is the effect of dilation on the perimeter of a triangle?

Tip: The ratio of areas after dilation is the square of the scale factor.