A triangle has corners at $(6 ,9 )$ , $(-2 ,-1 )$ , and $(1 ,-1 )$ . If the triangle is dilated by a factor of $1/3$ about point #(-5 ,-2 ), how far will its centroid move?

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Answer 1 · 2018-07-14T08:48:12

$color(blue)("Distance moved by centroid " color(crimson)(vec(GG') ~~ 5.3008 " units"$

$A(6,9), B(-2,-1), C(1, -1), " about point " D (-5, -2), " dilation factor "1/3$

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

$G(x,y) = ((6 - 2 + 1)/3, (9 - 1 - 1)/3) = (5/3, 7/3)$

$A'((x),(y)) = 1/3a - -2/3d = 1/3*((6),(9)) - -2/3*((-5),(-2)) = ((-4/3),(5/3))$

$B'((x),(y)) = 1/3b - -2/3d = 1/3*((-2),(-1)) - -2/3*((-5),(-2)) = ((-4),(-5/3))$

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

$"New Centroid " G'(x,y) = ((-4/3 - 4 - 3)/3,(5/3 - 5/3 - 5/3)/3) = (-25/9,-5/9)$

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

$color(crimson)(vec(GG') = sqrt((5/3- -25/9)^2 + (7/3 - -5/9)) ~~ 5.3008 " units"$

A triangle has corners at (6 ,9 )(6 ,9 ), (-2 ,-1 )(-2 ,-1 ), and (1 ,-1 )(1 ,-1 ). If the triangle is dilated by a factor of 1/3 1/3 about point #(-5 ,-2 ), how far will its centroid move?