Let's use the given coordinates to draw the original figure and the image of its dilation.
We can see that △ ABC and △ ADE share ∠ A, and we know that ∠ A ≅ ∠ A. Let's use the coordinates of the vertices to find the lengths of the sides of each triangle that include ∠ A.
Side
Vertices
Distance Formula
Simplified
AB
( 1,3), ( - 1,2)
sqrt(( - 1- 1)^2+( 2- 3)^2)
sqrt(5)
AD
( 1,3), (- 7,- 1)
sqrt((- 7- 1)^2+(- 1- 3)^2)
4sqrt(5)
AC
( 1,3), ( 1,1)
sqrt(( 1- 1)^2+( 1- 3)^2)
2
AE
( 1,3), ( 1,- 5)
sqrt(( 1- 1)^2+( - 5- 3)^2)
8
Now, we can find the ratios between the corresponding sides.
AD/AB=4sqrt(5)/sqrt(5) = 4 [1.2em]
AE/AC=8/2 = 4
We can tell that these ratios are equivalent. Therefore, the lengths of two sides of △ ABC are proportional to the lengths of two corresponding sides of △ ADE, and the included angles are congruent. By the Side-Angle-Side Similarity Theorem, we can conclude that △ ABC is similar to △ ADE.
△ ABC ~ △ ADE
Therefore, the dilation is a similarity transformation.
