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A triangle has corners at #(1 ,3 )#, #(2 ,-2 )#, and #(-8 ,6 )#. If the triangle is dilated by a factor of #5 # about point #(-3 ,2 ), how far will its centroid move?

Answer 1

Jayden Jackson

#color(green)("Distance moved by centroid " #

#color(blue)(vec(GG') ~~ 14.3604 " units"#

#A(1,3), B(2, -2), C(-8, 6), " about point " D (-3,2), " dilation factor "5#

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

#G(x,y) = ((1+2-8)/3, (3 - 2 + 6)/3) = (-5/3, -7/3)#

#A'((x),(y)) = 5a - 4d = 5*((1),(3)) - 4*((-3),(2)) = ((17),(7))#

#B'((x),(y)) = 5b - 4d = 5*((2),(-2)) - 4*((-3),(2)) = ((22), (-18))#

#C'((x),(y)) = 5c - 4d = 5*((-8),(6)) - 4*((-3),(2)) = ((-28),(-22))#

#"New Centroid " G'(x,y) = ((17+ 22 - 28)/3,(7 - 18 -22)/3) = (11/3, -11)#

#color(green)("Distance moved by centroid " #

#color(blue)(vec(GG') = sqrt((-5/3 - 11/3)^2 + (-7/3 - 11)) ~~ 14.3604 " units"#

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Answer 2

Sophia Alfred

The centroid of a triangle moves in the same direction and by the same factor as the dilation factor. So, if the triangle is dilated by a factor of 5, the centroid will move 5 times as far from its original position.

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