Given : #A (8,3), B (4,-6), C (-2,-4)# Dilated about #D(1,-3)# by a factor of 5.

To find the distance moved by centroid.

#vec(A’D) = 5 * vec(AD)#

#a’ - d = 5a - 5d# or #a’ = 5a - 4d#

#a’((x),(y)) = 5((8),(3)) - 4 ((1),(-3)) = ((40),(15)) - ((4),(-12)) = ((36),(27))#

Similarly,

#b’((x),(y)) = 5((4),(-6)) - 4 ((1),(-3)) = ((20),(-30)) - ((4),(-12)) = ((16),(18))#

#b’((x),(y)) = 5((-2),(-4)) - 4 ((1),(-3)) = ((-10),(-20)) - ((4),(-12)) = ((-6),(-8))#

Original centroid #g((x),(y)) = (((8+4-2)/3),((3-6-4)/3)) = ((10/3),(-7/3))#

New centroid after dilation

#g’((x),(y)) = (((36 + 16 -6)/3),((27+18-8)/3)) = ((46/3),(37/3))#

Distance moved by centroid

#vec(GG’) = sqrt((10/3-46/3)^2+ (-7/3 + 37/3)^2) = sqrt244 = color (green)(15.6205#