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

Question

1466 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-02-06T06:04:11

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

enter image source here

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

A triangle as corners at (5 ,2 )(5,2)(5 ,2 ), (2 ,4 )(2,4)(2 ,4 ), and (3 ,8 )(3,8)(3 ,8 ). If the triangle is dilated by a factor of 3 33 about #(2 ,3 ), how far will its centroid move?