A triangle has corners at #(-7 ,3 )#, #(4 ,-1 )#, and #(2 ,1 )#. If the triangle is dilated by a factor of #5 # about point #(1 ,-5 ), how far will its centroid move?

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Answer 1 · 2018-02-12T12:26:44

Centroid will move by #~~ color(green)(10.75)#

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

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

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

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

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

#=> 5((-7),(3)) - 4((1),(-5)) = ((-35),(15)) - ((4),(-20)) = ((-31),(-5))#

#color(blue)(A' (-31, -5)#

Similarly,

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

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

#=> 5((4),(-1)) - 4((1),(-5)) = ((20),(-5)) - ((4),(-20)) = ((16),(15))#

#color(blue)(B' (16, 15)#

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

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

#=> 5((2,(1)) - 4((1),(-5)) = ((10),(5)) - ((4),(-20)) = ((6),(25))#

#color(blue)(C' (6, 25)#

New centroid #G' = (-31 + 16+6)/3, (-5+15+25)/3 = color(brown)((-3, 35/3)#

Distance moved by centroid is

#vec(GG') = sqrt((-1/3-3)^2 + (1-(35/3))^2) ~~ color(green)(10.75)# rounded to two decimals