A triangle has corners at $(- 3, - 1)$ $(-3 ,-1 )$ , $(5, 6)$ $(5 ,6 )$ , and $(- 4, 7)$ $(-4 ,7 )$ . If the triangle is dilated by a factor of $\frac{1}{3}$ $1/3$ about point #(2 ,-1 ), how far will its centroid move?

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Answer 1 · 2018-07-14T08:38:37

$color(green)("Distance moved by centroid " color(crimson)(vec(GG') ~~ 4.4721 " units"$

$A(-3,-1), B(5,6), C(-4, 7), " about point " D (2,-1), " dilation factor "1/3$

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

$G(x,y) = ((-3+5 - 4)/3, (-1 + 6 + 7)/3) = (-2/3, 4)$

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

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

$C'((x),(y)) = 1/3c - -2/3d = 1/3*((-4),(7)) - -2/3*((2),(-1)) = ((0),(5/3))$

$"New Centroid " G'(x,y) = ((1/3+ 3 + 0)/3,(-1+ 4/3 + 5/3)/3) = (10/3,2)$

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

$color(crimson)(vec(GG') = sqrt((-2/3- 10/3)^2 + (4 - 2)) ~~ 4.4721 " units"$

A triangle has corners at (-3 ,-1 )(−3,−1)(-3 ,-1 ), (5 ,6 )(5,6)(5 ,6 ), and (-4 ,7 )(−4,7)(-4 ,7 ). If the triangle is dilated by a factor of 1/3 131/3 about point #(2 ,-1 ), how far will its centroid move?