Final answer:
The new endpoints of the dilated line segment with a dilation factor of 2 around (2, 4) are (-6, -8) and (-2, 4). The length of the dilated line segment is 4√10 units.
Step-by-step explanation:
When dilating a line segment by a factor of 2 about the point (2, 4), we effectively double the distance from this center of dilation to each endpoint of the line segment. To find the new endpoints, we first calculate the vector from the center of dilation to each original endpoint and then double that vector.
For the endpoint (-2, -2), the vector from the center (2, 4) is:
(-2 - 2, -2 - 4) = (-4, -6). Doubling this vector gives us (-8, -12). Adding this to the center of dilation gives the new endpoint:
(2 - 8, 4 - 12) = (-6, -8).
For the endpoint at (0, 4), the vector to the center is:
(0 - 2, 4 - 4) = (-2, 0). Doubling this gives (-4, 0), and adding it to the center of dilation provides the new endpoint:
(2 - 4, 4 + 0) = (-2, 4).
To find the length of the newly dilated line segment, we calculate the distance between the new endpoints (-6, -8) and (-2, 4) using the distance formula. The change in x-values is 4 (-2 - (-6)) and the change in y-values is 12 (4 - (-8)), so the distance is √(4² + 12²) = √160, which simplifies to 4√10.