A triangle has corners at #(-7 ,-2 )#, #(3 ,-1 )#, and #(5 ,7 )#. If the triangle is dilated by a factor of #2/5 # about point #(6 ,-8 ), how far will its centroid move?

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Answer 1 · 2018-02-12T12:46:22

Centroid has moved by a distance of #~~ color(green)(6.55)# units

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Given : A (-7,-2), B (3,-1), C(5,7)

Dilated about D(6,-8) and dilation factor 2/5

To find the distance moved by centroid.

Centroid #G = (-7+3+5)/3, (-2-1+7)/3 = color(brown)((1/3,4/3)#

#vec(A'D) = (2/5) * vec(AD)#

#a' - d = (2/5)(a - d)# or #a' = 0.4a + 0.6d#

#=> 0.4((-7),(-2)) + 0.6((6),(-8)) = ((-2.8),(-0.8)) + ((3.6,(-4.8)) = ((0.8),(-5.6))#

#color(blue)(A' (0.8, -5.6)#

Similarly,

#vec(B'D) = 5 * vec(BD)#

#b' - d = (2/5)(b - d)# or #b' = 0.4b + 0.6d#

#=> 0.4((3),(-1)) + 0.6((6),(-8)) = ((1.2),(-0.4)) + ((3.6,(-4.8)) = ((4.8),(-5.2))#

#color(blue)(B' (4.8, -5.2)#

#vec(C'D) = (2/5) * vec(CD)#

#c' - d = (2/5)(c - d)# or #c' = 0.4c + 0.6d#

#=> 0.4((5),(7)) + 0.6((6),(-8)) = ((2),(2.8)) + ((3.6,(-4.8)) = ((5.6),(-2))#

#color(blue)(C' (5.6, -2)#

New centroid #G' = (0.8 +4.8+5.6)/3, (-5.6-5.2-2)/3 = color(brown)((11.2/3, -12.8/3)#

Distance moved by centroid is

#vec(GG') = sqrt(((1/3)-11.2/3)^2 + ((4/3)+(12.8/3))^2) ~~ color(green)(6.55)# units