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?

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Answer 1 · 2018-04-30T11:42:19

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

#A(-1,7), B(3,-5), C(7,-3)# Dilated by 5 around #D(-2,4)#

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"#