In the given image, there is a dilation transformation of triangle $\triangle GEF$ to $\triangle G'E'F'$. To determine the scale factor used in this dilation, we need to compare the distances between corresponding points of the triangles.

Let's analyze the coordinates:

• Coordinates of $G$: $(-4, 2)$
• Coordinates of $G'$: $(-2, 1)$
• Coordinates of $E$: $(2, 5)$
• Coordinates of $E'$: $(1, 3)$
• Coordinates of $F$: $(4, 2)$
• Coordinates of $F'$: $(2, 1)$

Scale Factor Calculation:

The formula for the scale factor $k$ in dilation is:

$k = \frac{\text{Distance between corresponding points on the image and pre-image}}{\text{Original distance between corresponding points}}$

1. Distance between $G$ and $G'$: $\text{Distance} = \sqrt{(-2 - (-4))^2 + (1 - 2)^2} = \sqrt{(2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5}$

2. Distance between $E$ and $E'$: $\text{Distance} = \sqrt{(1 - 2)^2 + (3 - 5)^2} = \sqrt{(-1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5}$

So, all distances are halved compared to the original triangle, making the scale factor 1/2.

Thus, the scale factor used in this dilation is 1/2.

Would you like a deeper explanation or have any other questions?

Here are 5 questions related to this topic:

1. What is the general formula for dilation on the coordinate plane?
2. How does the center of dilation affect the transformed shape?
3. What happens to the orientation of a figure during a dilation?
4. How can you find the scale factor if you are only given the coordinates of the vertices?
5. What is the difference between dilation and other transformations like translation or rotation?

Tip: To find a dilation's scale factor, compare distances from the origin (if the dilation is centered at the origin) or between corresponding points.