A triangle has corners at $(2 ,-2 )$ , $(3 ,-1 )$ , and $(5 ,7 )$ . If the triangle is dilated by a factor of $2/5$ about point #(6 ,-2 ), how far will its centroid move?

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Answer 1 · 2018-07-14T09:06:58

$color(indigo)("Distance moved by centroid " color(magenta)(vec(GG') ~~ 3.3333" units"$

$A(2,-2), B(3, -1), C(5, 7), " about point " D (6,-2), " dilation factor "2/5$

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

$G(x,y) = ((2 + 3+ 5)/3, (-2 - 1 + 7)/3) = (10/3, 4/3)$

$A'((x),(y)) = 2/5a - -3/5d = 2/5*((2),(-2)) - -3/5*((6),(-2)) = ((22/5),(-2))$

$B'((x),(y)) = 2/5b - -3/5d = 2/5*((3),(-1)) - -3/5*((6),(-2)) = ((24/5),(-8/5))$

$C'((x),(y)) = 2/5c - -3/5d = 2/5*((5),(7)) - -3/5*((6),(-2)) = ((28/5),(8/5))$

$"New Centroid " G'(x,y) = ((22/5+ 24/5 + 28/5)/3,(-2 -8/5 + 8/5)/3) = (74/15,-2/3)$

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

$color(magenta)(vec(GG') = sqrt((10/3- 74/15)^2 + (4/3 - -2/3)) ~~ 3.3333" units"$

A triangle has corners at (2 ,-2 )(2 ,-2 ), (3 ,-1 )(3 ,-1 ), and (5 ,7 )(5 ,7 ). If the triangle is dilated by a factor of 2/5 2/5 about point #(6 ,-2 ), how far will its centroid move?