A triangle has corners at $(6, 3)$ $(6 ,3 )$ , $(4, - 1)$ $(4 ,-1 )$ , and $(2, - 9)$ $(2 ,-9 )$ . If the triangle is dilated by a factor of $5$ $5$ about point #(1 ,-5 ), how far will its centroid move?

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Answer 1 · 2018-07-14T08:28:41

$color(green)("Distance moved by centroid " color(purple)(vec(GG') ~~ 13.4164 " units"$

$A(6,3), B(4,-1), C(2,-9), " about point " D (1,-5), " dilation factor "5$

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

$G(x,y) = ((6+4+2)/3, (3 - 1 - 9)/3) = (4, -7/3)$

$A'((x),(y)) = 5a - 4d = 5*((6),(3)) - 4*((1),(-5)) = ((26),(35))$

$B'((x),(y)) = 5b - 4d = 5*((4),(-1)) - 4*(1),(-5)) = ((16),15))$

$C'((x),(y)) = 5c - 4d = 5*((2),(-9)) - 4*((1),(-5)) = ((6),(-25))$

$"New Centroid " G'(x,y) = ((26+ 16 + 6)/3,(35+ 15 -25)/3) = (16,25/3)$

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

$color(purple)(vec(GG') = sqrt((4- 16)^2 + (-7/3 + 25/3)) ~~ 13.4164 " units"$

A triangle has corners at (6 ,3 )(6,3)(6 ,3 ), (4 ,-1 )(4,−1)(4 ,-1 ), and (2 ,-9 )(2,−9)(2 ,-9 ). If the triangle is dilated by a factor of 5 55 about point #(1 ,-5 ), how far will its centroid move?