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

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Answer 1 · 2017-03-20T17:45:40

The distance is #=24#

Let #ABC# be the triangle

#A=(-1,7)#

#B=(-5,-3)#

#C=(2,9)#

The centroid of triangle #ABC# is

#C_c=((-1-5+2)/3,(7+(-3)+9)/3)=(4/3,13/3)#

Let #A'B'C'# be the triangle after the dilatation

The center of dilatation is #D=(-7,1)#

#vec(DA')=5vec(DA)=5*<6,6> = <30,30>#

#A'=(30-7,30+1)=(23,31)#

#vec(DB')=5vec(DB)=5*<2,-4> = <10,-20>#

#B'=(10-7,-20+1)=(3,-19)#

#vec(DC')=5vec(DC)=5*<9,8> = <45,40>#

#C'=(45-7,40+1)=(38,41)#

The centroid #C_c'# of triangle #A'B'C'# is

#C_c'=((23+3+38)/3,(31-19+41)/3)=(64/3,53/3)#

The distance between the 2 centroids is

#C_cC_c'=sqrt((64/3-4/3)^2+(53/3-13/3)^2)#

#=1/3sqrt(60^2+40^2)=72.11/3=24#