A triangle has corners at #(8 ,2 )#, #(4 ,9 )#, and #(7 ,3 )#. If the triangle is dilated by a factor of #5 # about point #(1 ,-6 ), how far will its centroid move?

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Answer 1 · 2017-03-17T16:05:29

THe distance between the centroids is #=47.7#

Let #ABC# be the triangle

#A=(8.2)#

#B=(4,9)#

#C=(7,3)#

The centroid of triangle #ABC# is

#C_c=((8+4+7)/3,(2+9+3)/3)=(19/3,14/3)#

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

The center of dilatation is #D=(1,-6)#

#vec(DA')=5vec(DA)=5*<7,8> = <35,40>#

#A'=(35+1,40-6)=(36,34)#

#vec(DB')=5vec(DB)=5*<3,15> = <15,75>#

#B'=(15+1,75-6)=(16,69)#

#vec(DC')=5vec(Dc)=5*<6,9> = <30,45>#

#C'=(30+1,45-6)=(31,39)#

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

#C_c'=((36+16+31)/3,(34+69+39)/3)=(83/3,142/3)#

The distance between the 2 centroids is

#C_cC_c'=sqrt((83/3-19/3)^2+(142/3-14/3)^2)#

#=1/3sqrt(64^2+128^2)=143.1/3=47.7#