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

Answer 1

Claire Akin

Centroid moves by #~~ color(green)(54.62)#

Centroid #G(((2+4-1)/3),((1-3+4)/3)) => ((-5/3),(2/3))#

Dilated about D(4-9) by a factor of 5

To find new coordinate of centroid G'

#vec(G'D) = 5 * vec(GD)#

#G'((x-4),(y+9)) = 5 * ((-5/3-4),(2/3+9))=> ((-85/3),(145/3))#

#G' ((-73/3),(118/3))#

#vec(GG') = sqrt((-5/3+85/3)^2 + (2/3-145/3)^2) ~~ color(green)(54.62)#

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Answer 2

Noah Atkinson

The centroid of a triangle moves proportionally with the scale factor of dilation. Given that the triangle is dilated by a factor of 5 about point (4, -9), the centroid will move 5 times the distance from the center of dilation to the original centroid. Thus, to find how far the centroid moves, we calculate the distance from the center of dilation (4, -9) to the original centroid, then multiply it by 5.

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