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

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Answer 1 · 2018-07-22T02:51:52

#color(purple)("Distance moved by centroid " color(crimson)(vec(GG') ~~ 0.4969 " units"#

#A(6, 2), B(-3, 4), C(1, -1), " about point " D (-2,-2), " 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) = ((6-3+1)/3, (2+4-1)/3) = (4/3, 5/3)#

#A'((x),(y)) = 1/3a - (2/3)d = (1/3)*((6),(2)) - (2/3)*((-2),(-2)) = ((10/3),(2))#

#B'((x),(y)) = (1/3)b - (2/3)d = (1/3)*((-3),(4)) - (2/3)*((-2),(-2)) = ((1/3),(8/3))#

#A'((x),(y)) = (1/3)c - (2/3)d = (1/3)*((1),(-1)) - (2/3)*((-2),(-2)) = ((5/3),(1))#

#"New Centroid " G'(x,y) = ((10/3+ 1/3+5/3)/3,(2+8/3+1)/3) = (16/9,17/9)#

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

#color(crimson)(vec(GG') = sqrt((16/9-4/3)^2 + (17/9-5/3)) ~~ 0.4969 " units"#