A triangle has corners at $(- 2, 1)$ $(-2 ,1 )$ , $(8, - 5)$ $(8 ,-5 )$ , and $(- 1, 4)$ $(-1 ,4 )$ . If the triangle is dilated by a factor of $5$ $5$ about point #(4 ,-6 ), how far will its centroid move?

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Answer 1 · 2018-06-08T01:50:23

$Centroid will by 25.75 units$ $color(crimson)("Centroid will by 25.75 units"$

$A (- 2, 1), b (8, - 5), C (- 1, 4), D (4, - 6) dilation factor 5$ $A (-2,1), b(8,-5), C(-1,4), D(4,-6) "dilation factor " 5$

Centroid $G = \frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3}$ $G = (x_1 + x_2 + x_3) /3, (y_1 + y_2 + y_3) / 3$

$G (x, y) = (\frac{- 2 + 8 - 1}{3}, \frac{1 - 5 + 4}{3}) = (\frac{5}{3}, 0)$ $G(x, y) = ((-2 + 8 - 1) / 3, (1 - 5 + 4) / 3) = (5/3, 0)$

$A' ((x),(y)) = 5a - 4d = 5*((-2),(1)) - 4*((4),(-6)) = ((-26), (29))$

$B' ((x),(y)) = 5b - 4d = 5*((8),(-5)) - 4*((4),(-6)) = ((24), (-1))$

$C' ((x),(y)) = 5c - 4d = 5*((-1),(4)) - 4*((4),(-6)) = ((-21), (44))$

$G'(x,y) = ((-26 + 24 -21) / 3, (29 - 1 + 44)/3) = (-23/3, 24)$

$bar(GG') = sqrt((-23/3-5/3)^2 + 24^2) = 25.75$ units

A triangle has corners at (-2 ,1 )(−2,1)(-2 ,1 ), (8 ,-5 )(8,−5)(8 ,-5 ), and (-1 ,4 )(−1,4)(-1 ,4 ). If the triangle is dilated by a factor of 5 55 about point #(4 ,-6 ), how far will its centroid move?