A triangle has corners at $(8 ,2 )$ , $(4 ,9 )$ , and $(-5 ,3 )$ . If the triangle is dilated by a factor of $5$ about point #(1 ,-2 ), how far will its centroid move?

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Answer 1 · 2018-02-12T11:46:51

Distance moved by centroid is $vec(GG') ~~ color(green)(28.83)$

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Given : A (8,2), B (4,9), C(-5,3)

Dilated about D(1,-2) and dilation factor 5

$Centroid$ G = (8+4+(-5))/3, (2+4+3)/3 = color(brown)((7/3,3)#

$vec(A'D) = 5 * vec(AD)$

$a' - d = 5(a - d)$ or $a' = 5a - 4d$

$=> 5((8),(2)) - 4((1),(-2)) = ((40),(10)) - ((4),(-8)) = ((36),(18))$

$color(blue)(A' (36, 18)$

$vec(B'D) = 5 * vec(BD)$

$b' - d = 5(b - d)$ or $b' = 5b - 4d$

$=> 5((4),(9)) - 4((1),(-2)) = ((20),(45)) - ((4),(-8)) = ((16),(53))$

$color(blue)(B' (16, 53)$

$vec(C'D) = 5 * vec(CD)$

$c' - D = 5(c - d)$ or $c' = 5c - 4d$

$=> 5((-5),(3)) - 4((1),(-2)) = ((-25),(15)) - ((4),(-8)) = ((-29),(23))$

$color(blue)(C' (-29, 23)$

New centroid $G' = (36 + 16-29)/3, (18+53+23)/3 = color(brown)((23/3, 94/3)$

Distance moved by centroid is

$vec(GG') = sqrt((7/3 - 23/3)^2 + (3 - 94/3)^2) ~~ color(green)(28.83)$

A triangle has corners at (8 ,2 )(8 ,2 ), (4 ,9 )(4 ,9 ), and (-5 ,3 )(-5 ,3 ). If the triangle is dilated by a factor of 5 5 about point #(1 ,-2 ), how far will its centroid move?