A triangle has corners at #(-3 ,-1 )#, #(7 ,3 )#, and #(1 ,2 )#. If the triangle is dilated by a factor of #1/3 # about point #(-3 ,4 ), how far will its centroid move?

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Answer 1 · 2018-07-14T08:55:15

#color(indigo)("Distance moved by centroid " color(crimson)(vec(GG') ~~ 3.02255 " units"#

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

#A'((x),(y)) = 1/3a - -2/3d = 1/3*((-3),(-1)) - -2/3*((-3),(4)) = ((-3),(7/3))#

#B'((x),(y)) = 1/3b - -2/3d = 1/3*((7),(3)) - -2/3*((-3),(4)) = ((1/3),(11/3))#

#C'((x),(y)) = 1/3c - -2/3d = 1/3*((1),(2)) - -2/3*((-3),(4)) = ((-5/3),(10/3))#

#"New Centroid " G'(x,y) = ((-3+ 1/3 - 5/3)/3,(7/3 + 11/3 + 10/3)/3) = (-7/9,28/9)#

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

#color(crimson)(vec(GG') = sqrt((5/3- -7/9)^2 + (4/3 -28/9)) ~~ 3.02255 " units"#