A triangle has corners at $(2, -5)$ , $(-8, 4)$ , and $(1, -3)$ . If the triangle is dilated by a factor of $2/5$ about point $(-1, -6)$ , how far will its centroid move?

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Answer 1 · 2018-07-14T01:41:10

$color(blue)(vec(GG') = sqrt((-5/3- -19/5)^2 + (-4/3- -76/15)) ~~ 3.7547 " units"$

$A(2,-5), B(-8,4), C(1,-3), " about point " D (-1,-6), " dilation factor "2/5$

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

$G(x,y) = ((2-8+1)/3, (-5+4-3)/3) = (-5/3, -4/3)$

$A'((x),(y)) = (2/5)a - (-3/5)d = (2/5)*((2),(-5)) + (3/5)*((-1),(-6)) = ((1/5),(-33/5))$

$B'((x),(y)) = (2/5)b - (-3/5)d = (2/5)*((-8),(4)) + (3/5)*((-1),(-6)) = ((-19/5),(-2))$

$A'((x),(y)) = (2/5)c - (-3/5)d = (2/5)*((1),(-3)) + (3/5)*((-1),(-6)) = ((-1/5),(-23/5))$

$"New centroid " G'(x,y) = ((1/5 - 19/5 - 1/5)/3,(-33/5 - 2 - 23/5) /3 = (-19/15,-76/15)$

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

$color(blue)(vec(GG') = sqrt((-5/3- -19/5)^2 + (-4/3- -76/15)) ~~ 3.7547 " units"$

A triangle has corners at (2, -5)(2, -5), (-8, 4)(-8, 4), and (1, -3)(1, -3). If the triangle is dilated by a factor of 2/52/5 about point (-1, -6)(-1, -6), how far will its centroid move?