A triangle has corners at #(-2 ,1 )#, #(6 ,-3 )#, and #(-1 ,4 )#. 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 · 2017-03-21T13:11:47

The distance is #=29#

Let #ABC# be the triangle

#A=(-2,1)#

#B=(6,-3)#

#C=(-1,4)#

The centroid of triangle #ABC# is

#C_c=((-2+6-1)/3,(1+(-3)+4)/3)=(1,2/3)#

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*<2,3> = <10,15>#

#B'=(10+4,15-6)=(14,9)#

#vec(DC')=5vec(DC)=5*<-5,10> = <-25,50>#

#C'=(-25+4,50-6)=(-19,44)#

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

#C_c'=((-26+14-19)/3,(29+9+44)/3)=(-31/3,82/3)#

The distance between the 2 centroids is

#C_cC_c'=sqrt((-31/3-1)^2+(82/3-2/3)^2)#

#=1/3sqrt(34^2+80^2)=86.93/3=29#