A triangle has corners at #(-1 ,3 )#, #(3 ,-2 )#, and #(8 ,4 )#. If the triangle is dilated by a factor of #5 # about point #(-2 ,6 ), how far will its centroid move?

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Answer 1 · 2018-02-05T09:13:35

Centroid moves by a distance of #~~color(green)(24.5)# units

Given A (-1,3), B(3,-2), C (8,4),

Dilation point D (-2,6), Dilation factor 5

To find
1. Present centroid,
2. centroid after dilation and
3. distance between the two centroids.

Present centroid #G = (-1 + 3 + 8)/3, (3 - 2 + 4) /3 = color(brown)((10/3, 5/3)#

#bar(DA') = 5 * (bar(DA))#

#a' - d = 5a - 5d#

#a' = 5a - 4d = 5((-1),(3)) - 4((-2),(6))#

#=> ((-5),(15)) - ((-8),(24)) = ((3),(-9))#

#color(green)(A' (3, -9)#

#bar(DB') = 5 * bar(DB)#

#b' = 5b - 4d = 5((3),(-2)) - 4 ((-2),(6))#

#=> ((15),(-10)) - ((-8),(24)) = ((23),(-34))#

#color(green)(B'(23, -34)#

Similarly, #bar(DC') = 5 * bar(DC)#

#c' = 5c - 4d = 5((8),(4)) - 4((-2),(6))#

#=> ((40),(20)) - ((-8),(24)) = ((48),(-4))#

#color(green)(C' (48, -4)#

Centroid of dilated triangle A'B'C' is

#G' = (3 + 23 + 48) / 3, (-9-34-4)/3 = color(brown)((74/3, -47/3)#

Distance between centroids can be found using distance formula,

#bar(GG') = sqrt(((74/3)-(10/3))^2 + ((-47/3) -(5/3))^2)#

#bar(GG') = sqrt((64/3)^2 + (52/3)^2) ~~ color(green)(24.5)#
(corrected to one decimal)