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

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Answer 1 · 2018-07-14T08:00:35

#color(crimson)(vec(GG') = sqrt((4/3-14/3)^2 + (7/3- -23/3)) ~~ 10.5409 " units"#

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

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

#G(x,y) = ((-2+1+5)/3, (3 - 4 - 6)/3) = (4/3, 7/3)#

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

#B'((x),(y)) = 2b - 1d = 2*((1),(-4)) - 1*(-2),(3)) = ((4),(-11))#

#C'((x),(y)) = 2c - 1d = 2*((5),(-6)) - 1*((-2),(3)) = ((12),(-15))#

#"New Centroid " G'(x,y) = ((-2+ 4+12)/3,(3- 11-15)/3) = (14/3,-23/3)#

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

#color(crimson)(vec(GG') = sqrt((4/3-14/3)^2 + (7/3- -23/3)) ~~ 10.5409 " units"#