A triangle has corners at $(8, 3)$ $(8 ,3 )$ , $(4, - 5)$ $(4 ,-5 )$ , and $(- 7, - 4)$ $(-7 ,-4 )$ . If the triangle is dilated by a factor of $5$ $5$ about point #(1 ,-3 ), how far will its centroid move?

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Answer 1 · 2018-02-10T01:53:05

Distance moved by centroid after dilation $color(green)(vec(GG') ~~ 11.39$

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Given : Points $A ((8),(3)), B ((4), (-5)), C ((-7),(-4))$ , $D ((1),(-3))$

Dilated by factor 5 about point $D ((1),(-3))$

To find the distance moved by G, the centroid

Present coordinates of centroid G

$G((x),(y)) -> (((8+4-7)/3), ((3-5-4)/3)) =>color(purple)( ((-5/3),(-2))$

Let $G'((x),(y))$ be the new coordinates of G

$vec(G'D) -> 5 * vec(GD)$

$((x-1),(y+3)) = 5 * (((-5/3)-1),((-2)+3))$

$x - 1 = 5 (-5/3 - 1)$

$3x - 3 = -25 - 15, 3x = -37, x = -37/3$

$y + 3 = 5 (-2 +3), y = 2$

$G' color(brown)(((-37/3),(2))$

$vec(G'G) = sqrt((-5/3+37/3)^2 + (-2 -2)^2) ~~ color(green)(11.39$

A triangle has corners at (8 ,3 )(8,3)(8 ,3 ), (4 ,-5 )(4,−5)(4 ,-5 ), and (-7 ,-4 )(−7,−4)(-7 ,-4 ). If the triangle is dilated by a factor of 5 55 about point #(1 ,-3 ), how far will its centroid move?