Let's use the given coordinates to draw the original figure and its dilated image.
We can see that △ GHJ and △ KHL share ∠ H, and we know that ∠ H ≅ ∠ H.
Let's use the coordinates of the vertices to find the lengths of the sides of each triangle that include ∠ H.
Side
Vertices
Distance Formula
Simplified
First Pair of Sides
HL
( -1,2), ( 1,0)
sqrt(( 1-( -1))^2+( 0- 2)^2)
sqrt(2) * sqrt(4)
HJ
( - 1,2), (2,- 1)
sqrt((2-( - 1))^2+(- 1- 2)^2)
sqrt(2) * sqrt(9)
Second Pair of Sides
HK
( -1,2), ( -3,-2)
sqrt(( - 3-( - 1))^2+( - 2- 2)^2)
sqrt(5) * sqrt(4)
HG
( -1,2), ( -4,-4)
sqrt(( -4-( -1))^2+( -4- 2)^2)
sqrt(5) * sqrt(9)
Now, we can find the ratios between the corresponding sides.
HL/HJ=sqrt(2) * sqrt(4)/sqrt(2) * sqrt(9) = sqrt(4)/sqrt(9) [1.2em]
HK/HG=sqrt(5) * sqrt(4)/sqrt(5) * sqrt(9) = sqrt(4)/sqrt(9)
We can tell that these ratios are equivalent. Therefore, the lengths of two sides of △ GHJ are proportional to the lengths of two corresponding sides of △ KHL, and the included angles are congruent. By the Side-Angle-Side Similarity Theorem, we can conclude that △ GHJ is similar to △ KHL.
△ GHJ ~ △ KHL
Therefore, the dilation is a similarity transformation.
