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

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Answer 1 · 2018-07-14T01:26:10

#color(green)(vec(GG') = sqrt((-2- -1/5)^2 + (1/3- -14/3)) ~~ 5.314 " units"#

#A(-7,-5), B(-2,-1), C(3,7), " about point " D (1,-8), " 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) = ((-7-2+3)/3, (-5-1+7)/3) = (-2, 1/3)#

#A'((x),(y)) = (2/5)a - (-3/5)d = (2/5)*((-7),(-5)) + (3/5)*((1),(-8)) = ((-11/5),(-34/5))#

#B'((x),(y)) = (2/5)b - (-3/5)d = (2/5)*((-2),(-1))+ (3/5)*((1),(-8)) = ((-1/5),(-26/5))#

#A'((x),(y)) = (2/5)c - (-3/5)d = (2/5)*((3),(7)) + (3/5)*((1),(-8)) = ((9/5),(-2))#

#"New centroid " G'(x,y) = ((-11/5 - 1/5 + 9/5)/3,(-34/5 - 26/5 - 2) /3 = (-1/5,-14/3)#

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

#color(green)(vec(GG') = sqrt((-2- -1/5)^2 + (1/3- -14/3)) ~~ 5.314 " units"#