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

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Answer 1 · 2018-07-14T08:21:03

#color(maroon)("Distance moved by centroid " color(indigo)(vec(GG') ~~ 3.6667 " units"#

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

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

#G(x,y) = ((5+9+2)/3, (3+ 4 + 2)/3) = (16/3, 3)#

#A'((x),(y)) = 2a - d = 2*((5),(3)) - 1*((9),(1)) = ((1),(5))#

#B'((x),(y)) = 2b - d = 2*((9),(4)) - 1*(9),(1)) = ((9),(7))#

#C'((x),(y)) = 2c - d = 2*((2),(2)) - 1*((9),(1)) = ((-5),(3))#

#"New Centroid " G'(x,y) = ((1+ 9- 5)/3,(5+ 7 -3)/3) = (5/3,3)#

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

#color(indigo)(vec(GG') = sqrt((16/3- 5/3)^2 + (3- 3)) ~~ 3.6667 " units"#