A triangle has corners at #(-5 ,6 )#, #(2 ,-3 )#, and #(8 ,9 )#. If the triangle is dilated by a factor of #5 # about point #(-3 ,6 ), how far will its centroid move?

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Answer 1 · 2018-02-20T08:58:53

Centroid will move by #color(brown)(23.409# units

#A(-5,6), B(2,-3), C(8,9)# Dilated by 5 around #D(-3,6)#

Centroid G(x,y) = (-5/3, 4)#

#A’ = 5A - 4D = 5((-5),(6)) - 4((-3),(6)) = ((-25),(30)) - ((-12),(16)) =color(blue)( ((-13),(14))#

#B’ = 5B - 4D = 5((2),(-3)) - 4((-3),(6)) = ((10),(-15)) - ((-12),(16)) =color(blue)( ((22),(-31))#

#C’ = 5C - 4D = 5((8),(9)) - 4((-3),(6)) = ((40),(35)) - ((-12),(16)) =color(blue)( ((52),(19))#

x Coordinate of new centroid =# G(x) = (a’ + b’ + c’)_x / 3 = (-13 + 22 + 52)/3 = 61/3#

y Coordinate of new centroid =# G(y) = (a’ + b’ + c’)_y / 3 = (14 - 31 + 19)/3 = 12#

Distance moved by centroid #d = sqrt((61/3 + 5/3)^2 + (12-4)^2) = color(brown)(23.4094#