A triangle has corners at $(4 ,-2 )$ , $(-1 ,2 )$ , and $(1 ,3 )$ . If the triangle is dilated by a factor of $1/3$ about point #(6 ,8 ), how far will its centroid move?

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Answer 1 · 2018-06-19T05:02:46

$color(crimson)(vec(GG') = sqrt((-32/9-4/3)^2 + (-1-1)) = 14.8 " units"$

$A(-2,6), B(4,-3), C(2,9), " about point " D (6,8), " dilation factor "1/3$

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

$G(x,y) = ((4-1+1)/3, (-2+2+3)/3) = (4/3, 1)$

$A'((x),(y)) = 1/3a - (2/3)d = (1/3)*((4),(-2)) - (2/3)*((6),(8)) = ((-8/3),(6))$

$B'((x),(y)) = (1/3)b - (2/3)d = (1/3)*((-1),(2)) - (2/3)*((6),(8)) = ((-13/3),(-14/3))$

$A'((x),(y)) = (1/3)c - (2/3)d = (1/3)*((1),(3)) - (2/3)*((6),(8)) = ((-11/3),(-13/3))$

$"New Centroid " G'(x,y) = ((-8/3-13/3-11/3)/3,(6-14/3-13/3)/3) = (-32/9,-1)$

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

$color(crimson)(vec(GG') = sqrt((-32/9-4/3)^2 + (-1-1)) = 14.8 " units"$

A triangle has corners at (4 ,-2 )(4 ,-2 ), (-1 ,2 )(-1 ,2 ), and (1 ,3 )(1 ,3 ). If the triangle is dilated by a factor of 1/3 1/3 about point #(6 ,8 ), how far will its centroid move?