A triangle has corners at #(2 ,2 )#, #(4 ,-7 )#, and #(3 ,4 )#. If the triangle is dilated by a factor of #5 # about point #(1 ,-9 ), how far will its centroid move?

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Answer 1 · 2017-03-30T06:15:33

The centroid will move by #=8.69#

Let #ABC# be the triangle

#A=(2,2)#

#B=(4,-7)#

#C=(3,4)#

The centroid of triangle #ABC# is

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

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

The center of dilatation is #D=(1,-9)#

#vec(DA')=5vec(DA)=5*<1,11> = <5,55>#

#A'=(5+1,55-9)=(6,46)#

#vec(DB')=5vec(DB)=5*<3,2> = <15,10>#

#B'=(15+1,10-9)=(16,1)#

#vec(DC')=5vec(DC)=5*<2,13> = <10,65>#

#C'=(10+1,65-9)=(11,56)#

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

#C_c'=((6+16+11)/3,(46+1+56)/3)=(11,103/3)#

The distance between the 2 centroids is

#C_cC_c'=sqrt((11-3)^2+(103/3+1/3)^2)#

#=1/3sqrt(64*9+104^2)=26.08/3=8.69#