Given : A (3,1), B (7,-5), C(-4,-2)

Dilated about D(7,-6) and dilation factor 5

To find distance the centroid has moved

#Centroid #G = (3+7+(-4))/3, (1-5-2)/3 = color(brown)((2,-2)#

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

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

#=> 5((3),(1)) - 4((-4),(-2)) = ((15),(5)) - ((-16),(-8)) = ((31),(13))#

#color(blue)(A' (31, 13)#

Similarly,

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

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

#=> 5((7),(-5)) - 4((-4),(-2)) = ((35),(-25)) - ((-16),(-8)) = ((51),(17))#

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

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

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

#=> 5((-4),(-2)) - 4((-4),(-2)) = ((-20),(-10)) - ((-16),(-8)) = ((-4),(-2))#

#color(blue)(C' (-4, -2)#

New centroid #G' = (31 + 51-4)/3, (13+17-2)/3 = color(brown)((26, 28/3)#

Distance moved by centroid is

#vec(GG') = sqrt((2-26)^2 + (-2-(28/3))^2) ~~ color(green)(26.54)# rounded to two decimals