A triangle has corners at #(-2 ,1 )#, #(8 ,-5 )#, and #(-1 ,-2 )#. If the triangle is dilated by a factor of #5 # about point #(4 ,-6 ), how far will its centroid move?

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Answer 1 · 2017-04-10T10:02:51

The centroid will move by #=18.52#

Let #ABC# be the triangle

#A=(-2,1)#

#B=(8,-5)#

#C=(-1,-2)#

The centroid of triangle #ABC# is

#C_c=((-2+8-1)/3,(1-5-2)/3)=(5/3,-2)#

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

The center of dilatation is #D=(4,-6)#

#vec(DA')=5vec(DA)=5*<-6,7> = <-30,35>#

#A'=(-30+4,35-6)=(-26,29)#

#vec(DB')=5vec(DB)=5*<4,1> = <20,5>#

#B'=(20+4,5-6)=(24,-1)#

#vec(DC')=5vec(DC)=5*<-5,4> = <-25,20>#

#C'=(-25+4,20-6)=(-21,14)#

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

#C_c'=((-26+24-21)/3,(29-1+14)/3)=(-23/3,42/3)#

The distance between the 2 centroids is

#C_cC_c'=sqrt((-23/3-5/3)^2+(42/3+2)^2)#

#=1/3sqrt(28^2+48^2)=55.57/3=18.52#