Let's take a look at the tasks from the image you uploaded:

Task 8:

It asks to find the scale factor of the dilation shown in the graph. This involves finding the relationship between the dimensions of the original and dilated figure.

By looking at the graph:

• The coordinates of the original figure are $D(0, 0), E(4, 0), F(4, 4), G(0, 4)$.
• The coordinates of the dilated figure are $D'(0, 0), E'(8, 0), F'(8, 8), G'(0, 8)$.

To find the scale factor $k$, we can compare the side lengths of the original figure and the dilated figure. For example, the side $DE$ in the original is 4 units, and the side $D'E'$ in the dilated figure is 8 units. Therefore, the scale factor is: $k = \frac{\text{new length}}{\text{original length}} = \frac{8}{4} = 2$ So the scale factor for this dilation is $k = 2$.

Task 9:

This task asks you to critique your friend's reasoning. They say the scale factor is $\frac{5}{2}$, but you are asked to find the correct scale factor. Looking at the graph:

• The original triangle has vertices $A(0, 0), B(2, -2), C(4, -4)$.
• The dilated triangle has vertices $A'(0, 0), B'(4, -4), C'(8, -8)$.

The scale factor $k$ is: $k = \frac{\text{new length}}{\text{original length}} = \frac{4}{2} = 2$ So the correct scale factor is 2. Your friend's mistake was likely miscalculating the ratio.

Task 10:

The task is asking for the scale factor of a dilation, where the smaller figure is the image of the larger one, and the origin is the center of dilation. From the graph:

• The original figure is a larger square with side length 12 units.
• The smaller figure has side length 6 units.

The scale factor $k$ is: $k = \frac{\text{new length}}{\text{original length}} = \frac{6}{12} = \frac{1}{2}$ So the scale factor is $\frac{1}{2}$, and this is a reduction.

Would you like a deeper explanation on any part of this? Or do you have other questions?

Here are some related questions you could explore:

1. How do scale factors affect the area of a figure in a dilation?
2. What happens to the scale factor when the dilation center is not at the origin?
3. How do you graph the dilated figures and the corresponding coordinates for a dilation?
4. How do you identify whether a dilation is an enlargement or reduction based on the scale factor?
5. What is the relationship between the scale factor and the perimeter of a dilated figure?

Tip: Always double-check the coordinates before calculating scale factors to avoid mistakes in dilation problems.