To determine the scale factor of the dilation between trapezoid $FGH\!I$ and $F'G'H'I'$, we can compare corresponding points between the original figure (green) and the dilated figure (red).

Steps:

1. Select a pair of corresponding points: Let's choose the points $F$ and $F'$, as well as $G$ and $G'$.

   • Coordinates of $F$ are approximately $(-6, 8)$, and coordinates of $F'$ are $(-3, 4)$.
   • Coordinates of $G$ are approximately $(0, 8)$, and coordinates of $G'$ are $(0, 4)$.

2. Find the distance between corresponding points and the origin:

   • For $F$ and $F'$:

     • Distance from $F$ to the origin: $\sqrt{(-6)^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$.
     • Distance from $F'$ to the origin: $\sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

   • For $G$ and $G'$:

     • Distance from $G$ to the origin: $\sqrt{0^2 + 8^2} = \sqrt{64} = 8$.
     • Distance from $G'$ to the origin: $\sqrt{0^2 + 4^2} = \sqrt{16} = 4$.

3. Calculate the scale factor:

   • The scale factor is the ratio of the distances from corresponding points. For example, using points $F$ and $F'$: $\text{Scale factor} = \frac{\text{Distance from origin to } F'}{\text{Distance from origin to } F} = \frac{5}{10} = \frac{1}{2}.$
   • Similarly, using points $G$ and $G'$: $\text{Scale factor} = \frac{4}{8} = \frac{1}{2}.$

Thus, the scale factor of the dilation is $\frac{1}{2}$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How do we verify dilation using other points like $H$ and $H'$?
2. Can we dilate figures with a scale factor greater than 1, and how does that change the figure?
3. What is the difference between dilation and translation in geometry?
4. How do you find the center of dilation from a given set of dilated points?
5. How does the concept of dilation apply in real-world scenarios?

Tip: When working with dilation, always ensure you use corresponding points and maintain proportional distances to confirm the scale factor.