# A triangle has corners at (-1 ,7 ), (3 ,-5 ), and (7 ,-3 ). If the triangle is dilated by a factor of 5  about point (-2 ,4 ), how far will its centroid move?

Apr 30, 2018

Distance moved by centroid d = color(brown)(12.67 " units"

#### Explanation:

$A \left(- 1 , 7\right) , B \left(3 , - 5\right) , C \left(7 , - 3\right)$ Dilated by 5 around $D \left(- 2 , 4\right)$

Centroid G(x,y) = (-1+3+7)/3, (7-5-3)/3 = (1, -1/3)

A’ = 5A - 4D = 5((-1),(7)) - 4((1),(-1/3)) = ((-5),(35)) - ((4),(-4/3)) =color(blue)( ((-9),(109/3))

B’ = 5B - 4D = 5((3),(-5)) - 4((1),(-1/3)) = ((15),(-25)) - ((4),(-4/3)) =color(blue)( ((11),(-71/3))

C’ = 5C - 4D = 5((7),(-3)) - 4((1),(-1/3)) = ((35),(-15)) - ((4),(-4/3)) =color(blue)( ((39),(-41/3))

x Coordinate of new centroid = G(x) = (a’ + b’ + c’)_x / 3 = (-9 + 11 + 39)/3 = 41/3

y Coordinate of new centroid = G(y) = (a’ + b’ + c’)_y / 3 = (109/3 - 71/3 - 41/3)/3 = -1/3

Distance moved by centroid d = sqrt((41/3 - 1)^2 + (-1/3 + 1/3)^2) = color(brown)(12.67 " units"