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

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Answer 1 · 2018-06-08T02:47:55

#color(green)("Centroid has moved by a distance of " 4.18 " units"#

#A (1,5), b(-1,-4), C(5,-6), D(-2,-3) "dilation factor " 2#

Centroid #G = (x_1 + x_2 + x_3) /3, (y_1 + y_2 + y_3) / 3#

#G(x, y) = ((1 - 1 + 5) / 3, (5 - 4 - 6) / 3) = (5/3, 5/3)#

#A' ((x),(y)) = 2a - d = 2*((1),(5)) - ((-2),(-3)) = ((4, 13)#

#B' ((x),(y)) = 2b - d = 2*((-1),(-4)) -( (-2),(-3)) = ((0, -5)#

#C' ((x),(y)) = 2c - d = 2*((5),(-6)) - ((-2),(-3)) = ((12, -9)#

#G'(x,y) = ((4 + 0 + 12) / 3, (13 - 5 - 9)/3) = (16/3, -1/3)#

#bar(GG') = sqrt((16/3-5/3)^2 + (-1/3-5/3)^2) = 4.18# units