Let's use the given coordinates to draw the original figure and its dilated image.
Let's use the coordinates of the vertices to find the lengths of the sides of each triangle.
Side
Vertices
Distance Formula
Simplified
First Pair of Sides
AC
(- 3,1), ( 3,- 2)
sqrt(( 3-(- 3))^2+( - 2-1)^2)
sqrt(45)
DF
( - 1,1), (1,0)
sqrt((1-( - 1))^2+( - 1)^2)
sqrt(5)
Second Pair of Sides
AB
(- 3,1), ( 9,7)
sqrt(( 9-(-3))^2+( 7-1)^2)
sqrt(180)
DE
( - 1,1), ( 3,3)
sqrt(( 3-( -1))^2+( 3- 1)^2)
sqrt(20)
Third Pair of Sides
CB
( 3,- 2), ( 9,7)
sqrt(( 9- 3)^2+( 7-( - 2))^2)
sqrt(117)
FE
(1,0), ( 3,3)
sqrt(( 3-1)^2+( 3- )^2)
sqrt(13)
Now, we can find the ratios between the corresponding sides.
AC/DF=sqrt(45)/sqrt(5) = 3 [1.2em]
AB/DE=sqrt(180)/sqrt(20) = 3 [1.2em]
CB/FE=sqrt(117)/sqrt(13) = 3
We can tell that these ratios are equivalent. Therefore, the corresponding side lengths of â–³ ABC and â–³ DEF are proportional. By the Side-Side-Side Similarity Theorem, we can conclude that â–³ ABC is similar to â–³ DEF.
â–³ ABC ~ â–³ DEF
Therefore, the dilation is a similarity transformation.
