A triangle has corners at #(6 ,9 )#, #(-2 ,-1 )#, and #(1 ,-1 )#. If the triangle is dilated by a factor of #1/3 # about point #(-5 ,-2 ), how far will its centroid move?

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Answer 1 · 2018-07-14T08:48:12

#color(blue)("Distance moved by centroid " color(crimson)(vec(GG') ~~ 5.3008 " units"#

#A(6,9), B(-2,-1), C(1, -1), " about point " D (-5, -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 - 2 + 1)/3, (9 - 1 - 1)/3) = (5/3, 7/3)#

#A'((x),(y)) = 1/3a - -2/3d = 1/3*((6),(9)) - -2/3*((-5),(-2)) = ((-4/3),(5/3))#

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

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

#"New Centroid " G'(x,y) = ((-4/3 - 4 - 3)/3,(5/3 - 5/3 - 5/3)/3) = (-25/9,-5/9)#

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

#color(crimson)(vec(GG') = sqrt((5/3- -25/9)^2 + (7/3 - -5/9)) ~~ 5.3008 " units"#