A triangle has corners at $(- 7, - 2)$ $(-7 ,-2 )$ , $(3, - 1)$ $(3 ,-1 )$ , and $(5, 7)$ $(5 ,7 )$ . If the triangle is dilated by a factor of $\frac{2}{5}$ $2/5$ about point #(6 ,-8 ), how far will its centroid move?

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Answer 1 · 2018-02-12T12:46:22

Centroid has moved by a distance of $\approx 6.55$ $~~ color(green)(6.55)$ units

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

Dilated about D(6,-8) and dilation factor 2/5

To find the distance moved by centroid.

Centroid $G = \frac{- 7 + 3 + 5}{3}, \frac{- 2 - 1 + 7}{3} = (\frac{1}{3}, \frac{4}{3})$ $G = (-7+3+5)/3, (-2-1+7)/3 = color(brown)((1/3,4/3)$

$vec(A'D) = (2/5) * vec(AD)$

$a' - d = (2/5)(a - d)$ or $a' = 0.4a + 0.6d$

$=> 0.4((-7),(-2)) + 0.6((6),(-8)) = ((-2.8),(-0.8)) + ((3.6,(-4.8)) = ((0.8),(-5.6))$

$color(blue)(A' (0.8, -5.6)$

Similarly,

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

$b' - d = (2/5)(b - d)$ or $b' = 0.4b + 0.6d$

$=> 0.4((3),(-1)) + 0.6((6),(-8)) = ((1.2),(-0.4)) + ((3.6,(-4.8)) = ((4.8),(-5.2))$

$color(blue)(B' (4.8, -5.2)$

$vec(C'D) = (2/5) * vec(CD)$

$c' - d = (2/5)(c - d)$ or $c' = 0.4c + 0.6d$

$=> 0.4((5),(7)) + 0.6((6),(-8)) = ((2),(2.8)) + ((3.6,(-4.8)) = ((5.6),(-2))$

$color(blue)(C' (5.6, -2)$

New centroid $G' = (0.8 +4.8+5.6)/3, (-5.6-5.2-2)/3 = color(brown)((11.2/3, -12.8/3)$

Distance moved by centroid is

$vec(GG') = sqrt(((1/3)-11.2/3)^2 + ((4/3)+(12.8/3))^2) ~~ color(green)(6.55)$ units

A triangle has corners at (-7 ,-2 )(−7,−2)(-7 ,-2 ), (3 ,-1 )(3,−1)(3 ,-1 ), and (5 ,7 )(5,7)(5 ,7 ). If the triangle is dilated by a factor of 2/5 252/5 about point #(6 ,-8 ), how far will its centroid move?