Question

A triangle as corners at `(5 ,2 )`, `(2 ,4 )`, and `(3 ,8 )`. If the triangle is dilated by a factor of `3` about #(2 ,3 ), how far will its centroid move?

Answer 1

centroid G has moved by a distance of `~~ color(green)(4.27` units rounded to 2 decimals

Explanation:

Given : A(5,2), B(2,4), C = (3,8), D(2,3)#

Dilated around D and by a factor of 3

To find the distance moved by the centroid `vec(GG')`

Coordinates of `G = (5 + 2 + 3)/3, (2+4+6)/3 = color(purple)((10/3, 14/3)`

`vec(A'D) = 3 * (vec(AD))`

`a' = 3a - 2d = 3((5),(2)) - 2((2),(3)) =color(blue)( ((11),(0))`

Similarly, `b' = 3b - 2d = 3((2),(4)) - 2((2),(3)) = color(blue)(((2),(6))`

and `c' = 3c - 2d = 3((3),(8)) - 2((2),(3)) = color(blue)(((5),(18))`

Coordinates of `G' = (11 + 2 + 5)/3, (0 + 6 + 18)/3 =color(purple)( (6,8)`

Using distance formula to find the distance between G & G',

`vec(GG') = sqrt((6-(10/3))^2 + (8-(14/3))^2) ~~ color(green)(4.27` rounded to 2 decimals