A triangle has corners at #(8 ,3 )#, #(4 ,-5 )#, and #(-7 ,-4 )#. If the triangle is dilated by a factor of #5 # about point #(1 ,-3 ), how far will its centroid move?

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Answer 1 · 2018-02-10T01:53:05

Distance moved by centroid after dilation #color(green)(vec(GG') ~~ 11.39#

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Given : Points #A ((8),(3)), B ((4), (-5)), C ((-7),(-4))#, #D ((1),(-3))#

Dilated by factor 5 about point #D ((1),(-3))#

To find the distance moved by G, the centroid

Present coordinates of centroid G

#G((x),(y)) -> (((8+4-7)/3), ((3-5-4)/3)) =>color(purple)( ((-5/3),(-2))#

Let #G'((x),(y))# be the new coordinates of G

#vec(G'D) -> 5 * vec(GD)#

#((x-1),(y+3)) = 5 * (((-5/3)-1),((-2)+3))#

#x - 1 = 5 (-5/3 - 1)#

#3x - 3 = -25 - 15, 3x = -37, x = -37/3#

#y + 3 = 5 (-2 +3), y = 2#

#G' color(brown)(((-37/3),(2))#

#vec(G'G) = sqrt((-5/3+37/3)^2 + (-2 -2)^2) ~~ color(green)(11.39#