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

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Answer 1 · 2018-06-08T01:50:23

#color(crimson)("Centroid will by 25.75 units"#

#A (-2,1), b(8,-5), C(-1,4), D(4,-6) "dilation factor " 5#

Centroid #G = (x_1 + x_2 + x_3) /3, (y_1 + y_2 + y_3) / 3#

#G(x, y) = ((-2 + 8 - 1) / 3, (1 - 5 + 4) / 3) = (5/3, 0)#

#A' ((x),(y)) = 5a - 4d = 5*((-2),(1)) - 4*((4),(-6)) = ((-26), (29))#

#B' ((x),(y)) = 5b - 4d = 5*((8),(-5)) - 4*((4),(-6)) = ((24), (-1))#

#C' ((x),(y)) = 5c - 4d = 5*((-1),(4)) - 4*((4),(-6)) = ((-21), (44))#

#G'(x,y) = ((-26 + 24 -21) / 3, (29 - 1 + 44)/3) = (-23/3, 24)#

#bar(GG') = sqrt((-23/3-5/3)^2 + 24^2) = 25.75# units