A triangle has corners at $(3 ,1 )$ , $(7 ,-5 )$ , and $(-4 ,-2 )$ . If the triangle is dilated by a factor of $5$ about point #(7 ,-6 ), how far will its centroid move?

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Answer 1 · 2018-02-05T11:19:58

Distance moved by centroid

$vec(GG') ~~ color(green)(26.54)$ rounded to two decimals

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Given : A (3,1), B (7,-5), C(-4,-2)

Dilated about D(7,-6) and dilation factor 5

To find distance the centroid has moved

$Centroid$ G = (3+7+(-4))/3, (1-5-2)/3 = color(brown)((2,-2)#

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

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

$=> 5((3),(1)) - 4((-4),(-2)) = ((15),(5)) - ((-16),(-8)) = ((31),(13))$

$color(blue)(A' (31, 13)$

Similarly,

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

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

$=> 5((7),(-5)) - 4((-4),(-2)) = ((35),(-25)) - ((-16),(-8)) = ((51),(17))$

$color(blue)(B' (51, 17)$

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

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

$=> 5((-4),(-2)) - 4((-4),(-2)) = ((-20),(-10)) - ((-16),(-8)) = ((-4),(-2))$

$color(blue)(C' (-4, -2)$

New centroid $G' = (31 + 51-4)/3, (13+17-2)/3 = color(brown)((26, 28/3)$

Distance moved by centroid is

$vec(GG') = sqrt((2-26)^2 + (-2-(28/3))^2) ~~ color(green)(26.54)$ rounded to two decimals

A triangle has corners at (3 ,1 )(3 ,1 ), (7 ,-5 )(7 ,-5 ), and (-4 ,-2 )(-4 ,-2 ). If the triangle is dilated by a factor of 5 5 about point #(7 ,-6 ), how far will its centroid move?