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

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Answer 1 · 2018-07-22T02:51:52

$color(purple)("Distance moved by centroid " color(crimson)(vec(GG') ~~ 0.4969 " units"$

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

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

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

$A'((x),(y)) = 1/3a - (2/3)d = (1/3)*((6),(2)) - (2/3)*((-2),(-2)) = ((10/3),(2))$

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

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

$"New Centroid " G'(x,y) = ((10/3+ 1/3+5/3)/3,(2+8/3+1)/3) = (16/9,17/9)$

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

$color(crimson)(vec(GG') = sqrt((16/9-4/3)^2 + (17/9-5/3)) ~~ 0.4969 " units"$

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