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

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Answer 1 · 2018-02-12T12:15:47

Centroid has moved by a distance of $\approx 26.36$ $~~ color(green)(26.36)$

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

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

To find the distance, centroid has moved

$C e n t r o i d$ $Centroid$ G = (1+3+(-1))/3, (3-2+7)/3 = color(brown)((1,8/3)#

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

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

$=> 5((1),(3)) - 4((-2),(-1)) = ((5),(15)) - ((-8),(-4)) = ((-3),(19))$

$color(blue)(A' (-3, 19)$

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

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

$=> 5((3),(-2)) - 4((-2),(-1)) = ((15),(-10)) - ((-8),(-4)) = ((51),(17))$

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

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

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

$=> 5((-1),(7)) - 4((-2),(-1)) = ((-5),(35)) - ((-8),(-4)) = ((-3),(39))$

$color(blue)(C' (-3, 39)$

New centroid $G' = (-3 + 51-3)/3, (19+17+39)/3 = color(brown)((15, 25)$

Distance moved by centroid is

$vec(GG') = sqrt((1-15)^2 + (8/3-25)^2) ~~ color(green)(26.36)$

A triangle has corners at (1, 3)(1,3)(1, 3), (3, -2)(3,−2)(3, -2), and (-1,7)(−1,7)(-1,7). If the triangle is dilated by a factor of 555 about point (-2, -1)(−2,−1)(-2, -1), how far will its centroid move?