Use the image to answer the question.
A coordinate grid of quadrant 1 shows 2 plotted triangles of the same shape but different sizes. The x and y axes range from 0 to 10 in unit increments. A small tilted triangle upper A upper B upper C is formed by connecting the following plots: point upper A is left parenthesis 2 comma 1 right parenthesis; point upper B is left parenthesis 3 comma 2 right parenthesis; and point upper C is left parenthesis 1 comma 3 right parenthesis. A large tilted dotted triangle is formed by connecting the following plots: point upper A prime is left parenthesis 6 comma 3 right parenthesis; point upper B prime is left parenthesis 9 comma 6 right parenthesis; point upper C prime is left parenthesis 3 comma 9 right parenthesis.
What is the scale factor of a single dilation between the preimage ABC and image A′B′C′?
(1 point)
The scale factor can be found by comparing the corresponding side lengths of the two triangles.
For the small triangle ABC, the lengths of the sides are:
AB = sqrt[(3-2)^2 + (2-1)^2] = sqrt[1+1] = sqrt(2)
BC = sqrt[(1-3)^2 + (3-2)^2] = sqrt[4+1] = sqrt(5)
AC = sqrt[(1-2)^2 + (3-1)^2] = sqrt[1+4] = sqrt(5)
For the large triangle A'B'C', the lengths of the sides are:
A'B' = sqrt[(9-6)^2 + (6-3)^2] = sqrt[9+9] = sqrt(18)
B'C' = sqrt[(3-9)^2 + (9-6)^2] = sqrt[36+9] = sqrt(45)
A'C' = sqrt[(3-6)^2 + (9-3)^2] = sqrt[9+36] = sqrt(45)
To find the scale factor, we can compare the ratios of the side lengths:
AB/A'B' = (sqrt(2))/(sqrt(18)) = sqrt(2/18) = sqrt(1/9) = 1/3
BC/B'C' = (sqrt(5))/(sqrt(45)) = sqrt(5/45) = sqrt(1/9) = 1/3
AC/A'C' = (sqrt(5))/(sqrt(45)) = sqrt(5/45) = sqrt(1/9) = 1/3
Since all the ratios are equal to 1/3, the scale factor of the dilation between the preimage ABC and image A'B'C' is 1/3.