A triangle has corners at #(4 ,-2 )#, #(-1 ,2 )#, and #(1 ,3 )#. If the triangle is dilated by a factor of #1/3 # about point #(6 ,8 ), how far will its centroid move?

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Answer 1 · 2018-06-19T05:02:46

#color(crimson)(vec(GG') = sqrt((-32/9-4/3)^2 + (-1-1)) = 14.8 " units"#

#A(-2,6), B(4,-3), C(2,9), " about point " D (6,8), " dilation factor "1/3#

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

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

#A'((x),(y)) = 1/3a - (2/3)d = (1/3)*((4),(-2)) - (2/3)*((6),(8)) = ((-8/3),(6))#

#B'((x),(y)) = (1/3)b - (2/3)d = (1/3)*((-1),(2)) - (2/3)*((6),(8)) = ((-13/3),(-14/3))#

#A'((x),(y)) = (1/3)c - (2/3)d = (1/3)*((1),(3)) - (2/3)*((6),(8)) = ((-11/3),(-13/3))#

#"New Centroid " G'(x,y) = ((-8/3-13/3-11/3)/3,(6-14/3-13/3)/3) = (-32/9,-1)#

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

#color(crimson)(vec(GG') = sqrt((-32/9-4/3)^2 + (-1-1)) = 14.8 " units"#