To find the vertices of the dilated triangle, we apply the given scale factor of 2 using the origin (0,0) as the center of dilation. The formula for dilation is:

$(x', y') = (k \cdot x, k \cdot y)$

where $k$ is the scale factor, and $(x, y)$ are the coordinates of the original point. In this case, $k = 2$, so each vertex will be multiplied by 2.

Original vertices:

1. $(-1, 4)$
2. $(-1, -2)$
3. $(2, -2)$

Now, apply the dilation to each vertex:

1. For $(-1, 4)$: $(-1 \cdot 2, 4 \cdot 2) = (-2, 8)$

2. For $(-1, -2)$: $(-1 \cdot 2, -2 \cdot 2) = (-2, -4)$

3. For $(2, -2)$: $(2 \cdot 2, -2 \cdot 2) = (4, -4)$

Dilated vertices:

• $(-2, 8)$
• $(-2, -4)$
• $(4, -4)$

Now, let's check the choices:

• A. (4, −2) → This does not match any dilated vertex.
• B. (−1, −4) → This does not match any dilated vertex.
• C. (4, −4) → This matches one of the dilated vertices.
• D. (−2, −4) → This matches one of the dilated vertices.
• E. (−2, 8) → This matches one of the dilated vertices.
• F. (−1, 4) → This is the original vertex, not dilated.

Final Answer:

The three dilated vertices are:

• C. (4, −4)
• D. (−2, −4)
• E. (−2, 8)

Would you like more details or have any further questions on this topic?

Here are five related questions:

1. How do you apply a dilation using a different center (not the origin)?
2. What is the effect of a scale factor of less than 1 on the dilation?
3. Can you describe the relationship between the original and dilated triangles visually?
4. How would the dilation of a square or rectangle look compared to the triangle in this case?
5. How do you handle dilations in different quadrants?

Tip: Dilation always involves scaling the distance from the center of dilation, so the figure maintains its shape but changes in size.