Final answer:
The transformation from ΔABC to ΔA'B'C' is confirmed as a dilation because the scale factor of 0.5 is constant for each pair of corresponding sides.
Step-by-step explanation:
To determine whether the transformation from ΔABC to ΔA'B'C' is a dilation, we need to compare the ratios of the distances between corresponding points and see if they are constant. Dilation is a type of transformation that produces an image that is the same shape as the original, but is a different size. It involves a scale factor which either enlarges or reduces the figure.
Let's consider the distances between corresponding points in the pre-image and the image:
- Distance between A and B in ΔABC is √((-3-0)^2+(4-4)^2) = √9 = 3 units.
- Distance between A' and B' in ΔA'B'C' is √((-1.5-0)^2+(2-2)^2) = √2.25 = 1.5 units.
The scale factor k from A to A' can be calculated as the ratio of A'B' to AB, which is 1.5/3 = 0.5. For this to be a dilation, the distances between all other corresponding points must also have the same scale factor.
- Distance between B and C in ΔABC is √((0-2)^2+(4-5)^2) = √5 = approximately 2.24 units.
- Distance between B' and C' in ΔA'B'C' is √((0-1)^2+(2-2.5)^2) = √1.25 = approximately 1.12 units.
The scale factor k from B to B' is also 1.12/2.24 = 0.5.
Since the scale factors for each pair of corresponding sides match, we can confirm that ΔABC has been dilated to ΔA'B'C' with a scale factor of 0.5.