To help with the dilation problem in the image, let's break it down.

Problem Overview:

• You are asked to perform a dilation on a line segment centered at the origin $(0, 0)$ with a scale factor of 2.
• After the dilation, you need to provide the coordinates of the endpoints of both the original figure and the final (dilated) figure.

Step 1: Identifying Original Coordinates

From the image, it seems that the original line segment has two points. The coordinates for these points need to be identified before dilation. Based on the graph:

• The left endpoint of the original figure is approximately $(2, 5)$.
• The right endpoint of the original figure is approximately $(6, 10)$.

Step 2: Applying the Dilation

To dilate a point $(x, y)$ by a scale factor $k$, multiply both coordinates by $k$: $(x', y') = (k \cdot x, k \cdot y)$ For a scale factor of 2:

• Left endpoint: $(2 \times 2, 5 \times 2) = (4, 10)$
• Right endpoint: $(6 \times 2, 10 \times 2) = (12, 20)$

Step 3: Fill in the Coordinates

Now, let's fill in the values:

Endpoints of original figure:

• Left: $(2, 5)$
• Right: $(6, 10)$

Endpoints of final (dilated) figure:

• Left: $(4, 10)$
• Right: $(12, 20)$

Would you like further clarification on any step, or is there anything else you'd like to explore?

Here are some follow-up questions you might find helpful:

1. How do you find the center of dilation if it's not at the origin?
2. How does the scale factor affect the length of a line segment?
3. What happens to the coordinates if the scale factor is less than 1?
4. How do you perform a dilation when the scale factor is negative?
5. How would you verify if the dilation transformation preserves angle measures?

Tip: Always check whether the dilation scale factor causes any points to move far beyond the graph's visible range.