A triangle has corners at (-7 ,-5 ), (-2 ,-1 ), and (3 ,7 ). If the triangle is dilated by a factor of 2/5 about point #(1 ,-8 ), how far will its centroid move?

1 Answer
Jul 14, 2018

color(green)(vec(GG') = sqrt((-2- -1/5)^2 + (1/3- -14/3)) ~~ 5.314 " units"

Explanation:

A(-7,-5), B(-2,-1), C(3,7), " about point " D (1,-8), " dilation factor "2/5

Centroid G(x,y) = ((x_a + x_b + x_c) /3, (y_a + y_b + y_c)/3)

G(x,y) = ((-7-2+3)/3, (-5-1+7)/3) = (-2, 1/3)

A'((x),(y)) = (2/5)a - (-3/5)d = (2/5)*((-7),(-5)) + (3/5)*((1),(-8)) = ((-11/5),(-34/5))

B'((x),(y)) = (2/5)b - (-3/5)d = (2/5)*((-2),(-1))+ (3/5)*((1),(-8)) = ((-1/5),(-26/5))

A'((x),(y)) = (2/5)c - (-3/5)d = (2/5)*((3),(7)) + (3/5)*((1),(-8)) = ((9/5),(-2))

"New centroid " G'(x,y) = ((-11/5 - 1/5 + 9/5)/3,(-34/5 - 26/5 - 2) /3 = (-1/5,-14/3)

color(green)("Distance moved by centroid "

color(green)(vec(GG') = sqrt((-2- -1/5)^2 + (1/3- -14/3)) ~~ 5.314 " units"