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A triangle has corners at #(6 ,3 )#, #(4 ,-1 )#, and #(3 ,-9 )#. If the triangle is dilated by a factor of #5 # about point #(8 ,-6 ), how far will its centroid move?

Answer 1

Sadie Adams

#"Distance moved by centroid is " = color(blue)(vec(GG') = 16.67 " units"#

#A(6,3), B(4,-1),C (3,-9), " about " D (8,-6), " with dilating factor " 5#

let G be the centroid.

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

#G(x,y) = G ((6 + 4 + 3) / 3, (3-1-9)/3) = G (13/3, -7/3)#

#A'((x),(y))= 5a - 4d = 5 * ((6),(3)) - 4 ((8),(-6)) = ((-2),(39))#

#B'((x),(y))= 5b - 4d = 5 * ((4),(-1)) - 4 ((8),(-6)) = ((-12),(19))#

#C'((x),(y))= 5c - 4d = 5 * ((3),(-9)) - 4 ((8),(-6)) = ((-17),(-21))#

#G'(x,y) = G'((-2-12-17)/3,(39+19-21)/3) = G'(-9,37/3)#

Distance moved by centroid is #vec(GG')#

#color(blue)(vec(GG') = sqrt((-9-13/3)^2 + (37/3-7/3)^2) = 16.67, " units"#

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Answer 2

Sadie Adams

The centroid of a triangle moves in the same ratio as the dilation factor. Therefore, if the triangle is dilated by a factor of 5, the centroid will move 5 times farther from the center of dilation compared to its original position.

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