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′ ?
To determine the scale factor of the dilation between the preimage ABC and the image A'B'C', we need to calculate the ratio between the corresponding side lengths of the two triangles.
First, let's find the side lengths of the preimage ABC:
Side AB: sqrt((3-2)^2+(2-1)^2) = sqrt(2^2+1^2) = sqrt(4+1) = sqrt(5)
Side BC: sqrt((1-3)^2+(3-2)^2) = sqrt((-2)^2+1^2) = sqrt(4+1) = sqrt(5)
Side AC: sqrt((3-1)^2+(2-3)^2) = sqrt(2^2+(-1)^2) = sqrt(4+1) = sqrt(5)
Now let's find the side lengths of the image A'B'C':
Side A'B': sqrt((9-6)^2+(6-3)^2) = sqrt(3^2+3^2) = sqrt(9+9) = sqrt(18) = 3*sqrt(2)
Side B'C': sqrt((3-9)^2+(9-6)^2) = sqrt((-6)^2+3^2) = sqrt(36+9) = sqrt(45) = 3*sqrt(5)
Side A'C': sqrt((6-3)^2+(3-9)^2) = sqrt(3^2+(-6)^2) = sqrt(9+36) = sqrt(45) = 3*sqrt(5)
Thus, the scale factor of the dilation is 3.