A triangle has corners at $(6, 3)$ $(6 ,3 )$ , $(4, - 1)$ $(4 ,-1 )$ , and $(3, - 9)$ $(3 ,-9 )$ . If the triangle is dilated by a factor of $5$ $5$ about point #(8 ,-6 ), how far will its centroid move?

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Answer 1 · 2018-06-08T10:05:35

$"Distance moved by centroid is " = color(blue)(vec(GG') = 16.67 " units"$

$A(6,3), B(4,-1),C (3,-9), " about " D (8,-6), " with dilating factor " 5$

let G be the centroid.

$G (x,y) = (x_a + x_b + x_c) / 3, (y_a + y_b + y_c)/3$

$G(x,y) = G ((6 + 4 + 3) / 3, (3-1-9)/3) = G (13/3, -7/3)$

$A'((x),(y))= 5a - 4d = 5 * ((6),(3)) - 4 ((8),(-6)) = ((-2),(39))$

$B'((x),(y))= 5b - 4d = 5 * ((4),(-1)) - 4 ((8),(-6)) = ((-12),(19))$

$C'((x),(y))= 5c - 4d = 5 * ((3),(-9)) - 4 ((8),(-6)) = ((-17),(-21))$

$G'(x,y) = G'((-2-12-17)/3,(39+19-21)/3) = G'(-9,37/3)$

Distance moved by centroid is $vec(GG')$

$color(blue)(vec(GG') = sqrt((-9-13/3)^2 + (37/3-7/3)^2) = 16.67, " units"$

A triangle has corners at (6 ,3 )(6,3)(6 ,3 ), (4 ,-1 )(4,−1)(4 ,-1 ), and (3 ,-9 )(3,−9)(3 ,-9 ). If the triangle is dilated by a factor of 5 55 about point #(8 ,-6 ), how far will its centroid move?