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

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Answer 1 · 2018-07-14T08:00:35

$color(crimson)(vec(GG') = sqrt((4/3-14/3)^2 + (7/3- -23/3)) ~~ 10.5409 " units"$

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

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

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

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

$B'((x),(y)) = 2b - 1d = 2*((1),(-4)) - 1*(-2),(3)) = ((4),(-11))$

$C'((x),(y)) = 2c - 1d = 2*((5),(-6)) - 1*((-2),(3)) = ((12),(-15))$

$"New Centroid " G'(x,y) = ((-2+ 4+12)/3,(3- 11-15)/3) = (14/3,-23/3)$

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

$color(crimson)(vec(GG') = sqrt((4/3-14/3)^2 + (7/3- -23/3)) ~~ 10.5409 " units"$

A triangle has corners at (-2 ,3 )(−2,3)(-2 ,3 ), (1 ,-4 )(1,−4)(1 ,-4 ), and (5 ,-6 )(5,−6)(5 ,-6 ). If the triangle is dilated by a factor of 2 22 about point #(-2 ,-3 ), how far will its centroid move?