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

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Answer 1 · 2018-06-07T10:57:49

#color(crimson)("Centroid will move by " 8.25 " units"#

#A (1,2), B (-3,-1), C (4,-2)#, dilated around #D(-5,7)# by a factor of 2.

#bar(A'D) = 2 * bar(AD)#

#A' - d = 2 * (a - d)#

#A' = 2a - d#

#A'((x),(y)) = 2 ((1),(2)) - ((-5),(7)) = ((7),(-3))#

Similarly,

#B' = 2b - d #

#B'((x),(y)) = 2 ((-3),(1)) - ((-5),(7)) = ((-1),(-5))#

#C' = 2c - d#

#C'((x),(y)) = 2 ((4),(-2)) - ((-5),(7)) = ((13),(-11))#

#"Centroid of Traingle ABC, " G= ((x_a + x_b + x_c)/3, (y_a + y_b + y_c) /3)#

#G(x,y) = ((1-3+4)/3, (2-1-2)/3) = (2/3, -1/3)#

#"Centroid of Traingle A'B'C', " G'= ((x_a' + x_b' + x_c')/3, (y_a' + y_b' + y_c') /3)#

#G'(x,y) = ((7-1+13)/3, (-3-5-11)/3) = (19/3, -19/3)#

#vec(GG') ' gives the distance moved by the centroid"#

#vec(GG') = sqrt((2/3 - 19/3)^2 + (-1/3 + 19/3)^2) = 8.25#