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

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Answer 1 · 2017-07-28T07:11:50

The centroid will move by #=18.1u#

Let the corners of the triangle be #(x_1,y_1)#, #(x_2,y_2)# and #(x_3,y_3)#

The coordinates of the centroid are

#C=((x_1+x_2+x_3)/2,(y_1+y_2+y_3)/3)#

Here, we have #(8,3)#, #(4,-5)#, and #(2,1)#

So,

The coordinates of the centroid are #C=((8+4+2)/3,(3-5+1)/3)=(14/3,-1/3)#

Let the coordinates of the centroid after dilatation be #C'=(x,y)#

The fixed point is #D=(1,-3)#

Therefore,

#vec(DC')=5vec(DC)#

#((x-1),(y+3))=5((14/3-1),(-1/3+3))=5((11/3),(8/3))=((55/3),(40/3))#

So,

#x-1=55/3#, #=>#, #x=55/3+1=58/3#

#y+3=40/3#, #=>#, #y=40/3-3=31/3#

So, the coordinates of #C'=(58/3,31/3)#

The distance between the centroids is

#C C'=sqrt((58/3-14/3)^2+(31/3+1/3)^2)#

#=sqrt((44/3)^2+(32/3)^2)#

#=sqrt(44^2+32^2)/3#

#=18.1u#