Question

A triangle has corners at `(4 ,-6 )`, `(3 ,-3 )`, and `(8 ,4 )`. If the triangle is dilated by a factor of `1/3` about point #(-3 ,2 ), how far will its centroid move?

Answer 1

Distance moved by centroid `~~ color(green)(2.53`

Explanation:

Formila for finding coordinates of centroid

`G(((x_1 + x_2 + x_3) /c),((y_1 + y_2 + y_3)/3)) = (((4+3+8)/3),((-6-3+4)/2)) = ((5),(-5/3))`

Dilated around point `((-3),(2))` by a factor of (1/3)#

New coordinates of centroid `vec(G'D)((x+3),(y-2)) = (1/3)vec(GD) = (1/3)((5+3),(-5/3-2)) => ((8/3),(-11/9))`

`G' ((17/3),(7/9))`

Distance between G & G'

`vec(GG') = sqrt((17/3-5)^2 + (7/9 + 5/3)^2) = sqrt((7/5)^2 + (4/9)^2) = ~~ color(green)(2.53`