A triangle has corners at #(2 ,-2 )#, #(3 ,-1 )#, and #(5 ,7 )#. If the triangle is dilated by a factor of #2/5 # about point #(6 ,-2 ), how far will its centroid move?

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Answer 1 · 2018-07-14T09:06:58

#color(indigo)("Distance moved by centroid " color(magenta)(vec(GG') ~~ 3.3333" units"#

#A(2,-2), B(3, -1), C(5, 7), " about point " D (6,-2), " 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 + 3+ 5)/3, (-2 - 1 + 7)/3) = (10/3, 4/3)#

#A'((x),(y)) = 2/5a - -3/5d = 2/5*((2),(-2)) - -3/5*((6),(-2)) = ((22/5),(-2))#

#B'((x),(y)) = 2/5b - -3/5d = 2/5*((3),(-1)) - -3/5*((6),(-2)) = ((24/5),(-8/5))#

#C'((x),(y)) = 2/5c - -3/5d = 2/5*((5),(7)) - -3/5*((6),(-2)) = ((28/5),(8/5))#

#"New Centroid " G'(x,y) = ((22/5+ 24/5 + 28/5)/3,(-2 -8/5 + 8/5)/3) = (74/15,-2/3)#

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

#color(magenta)(vec(GG') = sqrt((10/3- 74/15)^2 + (4/3 - -2/3)) ~~ 3.3333" units"#