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

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Answer 1 · 2018-07-22T01:49:22

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

$color(blue)(vec(GG') ~~ 14.3604 " units"$

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

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

$G(x,y) = ((1+2-8)/3, (3 - 2 + 6)/3) = (-5/3, -7/3)$

$A'((x),(y)) = 5a - 4d = 5*((1),(3)) - 4*((-3),(2)) = ((17),(7))$

$B'((x),(y)) = 5b - 4d = 5*((2),(-2)) - 4*((-3),(2)) = ((22), (-18))$

$C'((x),(y)) = 5c - 4d = 5*((-8),(6)) - 4*((-3),(2)) = ((-28),(-22))$

$"New Centroid " G'(x,y) = ((17+ 22 - 28)/3,(7 - 18 -22)/3) = (11/3, -11)$

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

$color(blue)(vec(GG') = sqrt((-5/3 - 11/3)^2 + (-7/3 - 11)) ~~ 14.3604 " units"$

A triangle has corners at (1 ,3 )(1,3)(1 ,3 ), (2 ,-2 )(2,−2)(2 ,-2 ), and (-8 ,6 )(−8,6)(-8 ,6 ). If the triangle is dilated by a factor of 5 55 about point #(-3 ,2 ), how far will its centroid move?