To determine the scale factor of the dilation from $DEFG$ to $D'E'F'G'$, we compare the lengths of corresponding sides of the two shapes. Let’s calculate this step by step:

Step 1: Identify corresponding sides

For simplicity, we will compare the side $EF$ from $DEFG$ with the corresponding side $E'F'$ from $D'E'F'G'$.

• Length of $EF$: The coordinates of $E$ are $(-4, -3)$ and $F$ are $(6, -3)$. The length of $EF$ is the horizontal distance: $EF = |6 - (-4)| = 6 + 4 = 10.$

• Length of $E'F'$: The coordinates of $E'$ are $(-2, -1.5)$ and $F'$ are $(3, -1.5)$. The length of $E'F'$ is the horizontal distance: $E'F' = |3 - (-2)| = 3 + 2 = 5.$

Step 2: Calculate the scale factor

The scale factor $k$ is given by the ratio of the length of the image ($E'F'$) to the original ($EF$): $k = \frac{E'F'}{EF} = \frac{5}{10} = \frac{1}{2}.$

Final Answer:

The scale factor is $\frac{1}{2}$. The correct option is C.

Would you like a detailed explanation of why this approach works or have additional questions?

Related Questions:

1. What happens to the area of the figure when it is dilated by a scale factor?
2. How can you determine the center of dilation for a transformation like this?
3. How do negative scale factors affect a dilation?
4. What other pairs of corresponding sides could be used to confirm the scale factor?
5. How does dilation differ from other transformations like reflection or rotation?

Tip:

Always ensure the coordinates of corresponding points are carefully calculated, as even small errors can affect the scale factor.