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

1 Answer
Jun 8, 2018

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

Explanation:

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"