A triangle has corners at #(8 ,2 )#, #(4 ,9 )#, and #(-5 ,3 )#. If the triangle is dilated by a factor of #5 # about point #(1 ,-2 ), how far will its centroid move?

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Answer 1 · 2018-02-12T11:46:51

Distance moved by centroid is #vec(GG') ~~ color(green)(28.83)#

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Given : A (8,2), B (4,9), C(-5,3)

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

#Centroid #G = (8+4+(-5))/3, (2+4+3)/3 = color(brown)((7/3,3)#

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

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

#=> 5((8),(2)) - 4((1),(-2)) = ((40),(10)) - ((4),(-8)) = ((36),(18))#

#color(blue)(A' (36, 18)#

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

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

#=> 5((4),(9)) - 4((1),(-2)) = ((20),(45)) - ((4),(-8)) = ((16),(53))#

#color(blue)(B' (16, 53)#

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

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

#=> 5((-5),(3)) - 4((1),(-2)) = ((-25),(15)) - ((4),(-8)) = ((-29),(23))#

#color(blue)(C' (-29, 23)#

New centroid #G' = (36 + 16-29)/3, (18+53+23)/3 = color(brown)((23/3, 94/3)#

Distance moved by centroid is

#vec(GG') = sqrt((7/3 - 23/3)^2 + (3 - 94/3)^2) ~~ color(green)(28.83)#