Let's use the given coordinates to draw the original figure and its dilated image.
We can see that △ SQR and △ PMN have right angles, ∠ Q and ∠ M, and therefore we know that ∠ Q ≅ ∠ M.
Let's use the coordinates of the vertices to find the lengths of the sides of each triangle that include the right angles.
Side
Vertices
Distance Formula
Simplified
First Pair of Sides
QS
( 2,-8), ( 2,- 2)
sqrt(( 2- 2)^2+( - 2-( - 8))^2)
6
MP
( 7,- 4), (7,- 1)
sqrt((7- 7)^2+(- 1-( - 4))^2)
3
Second Pair of Sides
QR
( 2,- 8), ( 6,- 8)
sqrt(( 6- 2)^2+( - 8-( - 8))^2)
4
MN
( 7,- 4), ( 5,-4)
sqrt(( 5- 7)^2+( -4-( - 4))^2)
2
Now, we can find the ratios between the corresponding sides.
QS/MP=6/3 = 2 [1.2em]
QR/MN=4/2 = 2
We can tell that these ratios are equivalent. Therefore, the lengths of two sides of △ SQR are proportional to the lengths of two corresponding sides of △ PMN, and the included angles are congruent. By the Side-Angle-Side Similarity Theorem, we can conclude that △ SQR is similar to △ PMN.
△ SQR ~ △ PMN
Therefore, the dilation is a similarity transformation.
