A triangle has corners at (2 ,-5 ), (-1 ,2 ), and (3 ,-3 ). If the triangle is dilated by a factor of 2/5  about point (-1 ,3 ), how far will its centroid move?

Feb 16, 2018

Distance moved by centroid is

vec(GG’) ~~ color (green)(2.5

Explanation:

Coordinates of centroid is obtained using the formula,

$G \left(x , y\right) = G \left(\frac{x 1 + x 2 + x 3}{3} , \frac{y 1 + y 2 + y 3}{3}\right)$

Given : $A \left(2 , - 5\right) , B \left(- 1 , 2\right) , C \left(3 , - 3\right)$

Centroid G ((x),(y)) = (((2-1+3)/3),((-5+2-3)/3)) = color(brown)(((4/3),(-2))

Triangle is dilated about point D(-1,3) by a factor of 2/5

bar(A’D) = (2/5) bar(AD)

a’ - d = (2/5) (a - d) = 0.4 a - 0.4d

a’((x),(y)) = 0.4((2),(-5)) + 0.6 ((-1),(2)) = color(blue)(((0.2),(-0.8))

b’((x),(y)) = 0.4((-1),(2)) + 0.6 ((-1),(2)) = color(blue)(((1),(2))

c’((x),(y)) = 0.4((3),(-3)) + 0.6 ((-1),(2)) = color(blue)(((0.6),(0))

New centroid G’((x),(y)) = (((0.2+1+0.6)/3),((-0.8+2+0)/3)) = color(brown)(((0.6),(0.4))

Using distance formula, we can calculate the distance moved by centroid

vec(GG’) = sqrt((4/3 - 0.6)^2 + (0.4+2)^2) ~~ color (green)(2.5#