To determine which dilation of [tex]\(\triangle RST\)[/tex] would result in a line segment with a slope of 2 that passes through [tex]\((-4, 2)\)[/tex], let's consider the properties of dilation and what it does to geometric figures.
A dilation transforms every point in a figure by the same factor from a center point. The slope of a line (which describes its steepness) remains unchanged under dilation, regardless of the scale factor, but the position of the line segments can shift based on the center of dilation.
We need to ensure two things:
1. The line segment that results from the dilation must have a slope of 2.
2. It must pass through the point [tex]\((-4, 2)\)[/tex].
Given the options, let's evaluate each scenario based on these criteria:
Option A:
- A dilation with a scale factor of 6 centered at [tex]\((-4, 2)\)[/tex].
If the center of dilation is [tex]\((-4, 2)\)[/tex], which is a point through which our line must pass, the dilation will scale the entire figure without changing the relative positions of points concerning this center. Consequently, any line passing through this point before dilation will still pass through this point after dilation. This ensures the resulting line can have a slope of 2 and still pass through [tex]\((-4, 2)\)[/tex].
Option B:
- A dilation with a scale factor of 0.5 centered at [tex]\((2, -4)\)[/tex].
In this case, the center of dilation is [tex]\((2, -4)\)[/tex], which is not [tex]\((-4, 2)\)[/tex]. Any line passing through [tex]\((-4, 2)\)[/tex] initially would shift its position owing to the dilation from a different center, thus failing to guarantee that it still passes through [tex]\((-4, 2)\)[/tex].
Option C:
- A dilation with a scale factor of 4 centered at [tex]\((0, -3)\)[/tex].
Similarly, with the center of dilation at [tex]\((0, -3)\)[/tex], the resulting distances and positions from this new center to other points would not ensure the line still goes through [tex]\((-4, 2)\)[/tex].
Option D:
- A dilation with a scale factor of 2 centered at [tex]\((3, 0)\)[/tex].
Again, as in the previous two options, this center of dilation [tex]\((3,0)\)[/tex] is not [tex]\((-4, 2)\)[/tex]. Any initial line passing through [tex]\((-4, 2)\)[/tex] might not still pass through the same point after dilation from a different center.
Given the conditions provided, only Option A meets both requirements.
Therefore, the correct answer is:
A. A dilation with a scale factor of 6 centered at [tex]\((-4, 2)\)[/tex].
