A triangle as corners at #(2 ,7 )#, #(5 ,8 )#, and #(3 ,1 )#. If the triangle is dilated by a factor of #3 # about #(6 ,5 ), how far will its centroid move?

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Answer 1 · 2018-07-14T08:15:12

#color(maroon)("Distance moved by centroid " color(green)(vec(GG') ~~ 1.4907 " units"#

#A(2,7), B(5,8), C(3,1), " about point " D (6,5), " dilation factor "3#

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

#G(x,y) = ((2+5+3)/3, (7+ 8 + 1)/3) = (10/3, -16/3)#

#A'((x),(y)) = 3a - 2d = 3*((2),(7)) - 2*((6),(5)) = ((-6),(11))#

#B'((x),(y)) = 3b - 2d = 3*((5),(8)) - 2*(6),(5)) = ((3),(14))#

#C'((x),(y)) = 3c - 2d = 3*((3),(1)) - 2*((6),(5)) = ((-3),(-7))#

#"New Centroid " G'(x,y) = ((-6+ 3- 3)/3,(11+ 14 -7)/3) = (-2,6)#

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

#color(green)(vec(GG') = sqrt((10/3- -2)^2 + (-16/3- 6)) ~~ 1.4907 " units"#