A triangle has corners at #(2, -5)#, #(-8, 4)#, and #(1, -3)#. If the triangle is dilated by a factor of #2/5# about point #(-1, -6)#, how far will its centroid move?

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Answer 1 · 2018-07-14T01:41:10

#color(blue)(vec(GG') = sqrt((-5/3- -19/5)^2 + (-4/3- -76/15)) ~~ 3.7547 " units"#

#A(2,-5), B(-8,4), C(1,-3), " about point " D (-1,-6), " dilation factor "2/5#

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

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

#A'((x),(y)) = (2/5)a - (-3/5)d = (2/5)*((2),(-5)) + (3/5)*((-1),(-6)) = ((1/5),(-33/5))#

#B'((x),(y)) = (2/5)b - (-3/5)d = (2/5)*((-8),(4)) + (3/5)*((-1),(-6)) = ((-19/5),(-2))#

#A'((x),(y)) = (2/5)c - (-3/5)d = (2/5)*((1),(-3)) + (3/5)*((-1),(-6)) = ((-1/5),(-23/5))#

#"New centroid " G'(x,y) = ((1/5 - 19/5 - 1/5)/3,(-33/5 - 2 - 23/5) /3 = (-19/15,-76/15)#

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

#color(blue)(vec(GG') = sqrt((-5/3- -19/5)^2 + (-4/3- -76/15)) ~~ 3.7547 " units"#