A triangle has corners at #(-2 ,1 )#, #(6 ,-3 )#, and #(-1 ,4 )#. If the triangle is dilated by a factor of #5 # about point #(4 ,-9 ), how far will its centroid move?

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Answer 1 · 2018-07-14T00:53:31

#color(crimson)(vec(GG') = sqrt((-1/3-1)^2 + (118/3-2/3)) ~~ 38.69 " units"#

#A(-2,1), B(6,-3), C(-1,4), " about point " D (4,-9), " dilation factor "5#

Centroid #G(x,y) = ((x_a + x_b + x_c) /3, (y_a + y_b + y_c)/3)#

#G(x,y) = ((-2+6-1)/3, (1 - 3 + 4)/3) = (1, 2/3)#

#A'((x),(y)) = 5a - 4d = 5*((-2),(1)) - 4*((4),(-9)) = ((-26),(41))#

#B'((x),(y)) = 5b - 4d = 5*((6),(-3)) - 4*((4),(-9)) = ((14),(21))#

#C'((x),(y)) = 5c - 4d = 5*((-1),(4)) - 4*((4),(-9)) = ((11),(56))#

#"New Centroid " G'(x,y) = ((-26+14+11)/3,(41+21+56)/3) = (-1/3,118/3)#

#color(purple)("Distance moved by centroid " #

#color(crimson)(vec(GG') = sqrt((-1/3-1)^2 + (118/3-2/3)) ~~ 38.69 " units"#