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

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Answer 1 · 2017-03-14T19:29:38

The distance is #=5.83#

Let #ABC# be the triangle

#A=(5.3)#

#B=(1,4)#

#C=(3.5)#

The centroid of triangle #ABC# is

#C_c=((5+1+3)/3,(3+4+5)/3)=(3,4)#

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

The center of dilatation is #D=(2,2)#

#vec(DA')=3vec(DA)=3*<3,1> = <9,3>#

#A'=(9+2,3+2)=(11,5)#

#vec(DB')=3vec(DB)=3*<-1,2> = <-3,6>#

#B'=(-3+2,6+2)=(-1,11)#

#vec(DC')=3vec(Dc)=3*<2,3> = <6,9>#

#C'=(6+2,9+2)=(8,11)#

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

#C_c'=((11-1+8)/3,(5+11+11)/3)=(6,9)#

The distance between the 2 centroids is

#C_cC_c'=sqrt((6-3)^2+(9-4)^2)#

#=sqrt(9+25)=sqrt34=5.83#