A triangle has corners at $(- 7, 3)$ $(-7 ,3 )$ , $(4, - 1)$ $(4 ,-1 )$ , and $(2, 1)$ $(2 ,1 )$ . 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 · 2018-02-12T12:26:44

Centroid will move by $\approx 10.75$ $~~ color(green)(10.75)$

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

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

$C e n t r o i d$ $Centroid$ G = (-7+4+2)/3, (3-1+1)/3 = color(brown)((-1/3,1)#

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

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

$=> 5((-7),(3)) - 4((1),(-5)) = ((-35),(15)) - ((4),(-20)) = ((-31),(-5))$

$color(blue)(A' (-31, -5)$

Similarly,

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

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

$=> 5((4),(-1)) - 4((1),(-5)) = ((20),(-5)) - ((4),(-20)) = ((16),(15))$

$color(blue)(B' (16, 15)$

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

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

$=> 5((2,(1)) - 4((1),(-5)) = ((10),(5)) - ((4),(-20)) = ((6),(25))$

$color(blue)(C' (6, 25)$

New centroid $G' = (-31 + 16+6)/3, (-5+15+25)/3 = color(brown)((-3, 35/3)$

Distance moved by centroid is

$vec(GG') = sqrt((-1/3-3)^2 + (1-(35/3))^2) ~~ color(green)(10.75)$ rounded to two decimals

A triangle has corners at (-7 ,3 )(−7,3)(-7 ,3 ), (4 ,-1 )(4,−1)(4 ,-1 ), and (2 ,1 )(2,1)(2 ,1 ). If the triangle is dilated by a factor of 5 55 about point #(1 ,-5 ), how far will its centroid move?