Given : A (1,3), B (3,-2), C(-1,7)

Dilated about D(-2,-1) and dilation factor 5

To find the distance, centroid has moved

#Centroid #G = (1+3+(-1))/3, (3-2+7)/3 = color(brown)((1,8/3)#

#vec(A'D) = 5 * vec(AD)#

#a' - d = 5(a - d)# or #a' = 5a - 4d#

#=> 5((1),(3)) - 4((-2),(-1)) = ((5),(15)) - ((-8),(-4)) = ((-3),(19))#

#color(blue)(A' (-3, 19)#

#vec(B'D) = 5 * vec(BD)#

#b' - d = 5(b - d)# or #b' = 5b - 4d#

#=> 5((3),(-2)) - 4((-2),(-1)) = ((15),(-10)) - ((-8),(-4)) = ((51),(17))#

#color(blue)(B' (51, 17)#

#vec(C'D) = 5 * vec(CD)#

#c' - D = 5(c - d)# or #c' = 5c - 4d#

#=> 5((-1),(7)) - 4((-2),(-1)) = ((-5),(35)) - ((-8),(-4)) = ((-3),(39))#

#color(blue)(C' (-3, 39)#

New centroid #G' = (-3 + 51-3)/3, (19+17+39)/3 = color(brown)((15, 25)#

Distance moved by centroid is

#vec(GG') = sqrt((1-15)^2 + (8/3-25)^2) ~~ color(green)(26.36)#