A triangle has corners at #(7 ,3 )#, #(9 ,4 )#, and #(5 ,2 )#. If the triangle is dilated by a factor of #2 # about point #(6 ,1 ), how far will its centroid move?

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Answer 1 · 2018-07-14T01:02:18

#color(purple)(vec(GG') = sqrt((22/3-7)^2 + (5-3)) ~~ 2.0276 " units"#

#A(7,3), B(9,4), C(5,2), " about point " D (6,1), " dilation factor "2#

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

#G(x,y) = ((7+9+5)/3, (3+4+2)/3) = (7, 3)#

#A'((x),(y)) = 2a - d = 2*((7),(3)) - ((6),(1)) = ((8),(5))#

#B'((x),(y)) = 2b - d = 2*((8),(4)) - ((6),(1)) = ((10),(7))#

#C'((x),(y)) = 2c - d = 2*((5),(2)) - ((6),(1) = ((4),(3))#

#"New Centroid " G'(x,y) = ((8+10+4)/3,(5+7+3)/3) = (22/3,5)#

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

#color(purple)(vec(GG') = sqrt((22/3-7)^2 + (5-3)) ~~ 2.0276 " units"#