Question

a triangle △abc has vertices a(1,1), b(3,1), and c(2,4). after a transformation, the vertices become a(2,2), b(6,2), and c(4,8). is the resulting figure similar, and what transformation was performed? a. no, the figures are not similar; a rotation was performed. b. yes, the figures are similar; a dilation was performed. c. yes, the figures are similar; a reflection was performed. d. no, the figures are not similar; a translation was performed.

Explanation:

Step1: Check if figures are similar

To check if two figures are similar, we look at the ratios of corresponding side - lengths. For a dilation, the ratio of corresponding coordinates of the pre - image and image points should be the same.
Let's find the ratios of the x - coordinates and y - coordinates of the corresponding points.
For point A(1,1) and A'(2,2):
The ratio of x - coordinates is $\frac{2}{1}=2$ and the ratio of y - coordinates is $\frac{2}{1}=2$.
For point B(3,1) and B'(6,2):
The ratio of x - coordinates is $\frac{6}{3}=2$ and the ratio of y - coordinates is $\frac{2}{1}=2$.
For point C(2,4) and C'(4,8):
The ratio of x - coordinates is $\frac{4}{2}=2$ and the ratio of y - coordinates is $\frac{8}{4}=2$.
Since the ratios of the corresponding coordinates of all pairs of corresponding points are equal, a dilation was performed and the figures are similar.

Answer:

b. Yes, the figures are similar; a dilation was performed.