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

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Answer 1 · 2018-04-26T11:52:08

Given vertices #(a,b),(c,d),(e,f)#, dilation point #(p,q)#, dilation factor #r#, the distance the centroid moves after dilation is:

# {r-1}/3 sqrt{ (a+c+e-3p)^2 + (b+d+e-3q)^2 }#

# = 4/3 sqrt( (-1+3+7-3(-2))^2 + (2+-5+4-3(6))^2 } = 4/3 sqrt{514}#

I've answered one or two of these before. Let's do this one in general:

Given a triangle with vertices #(a,b),(c,d),(e,f)# and dilation point #(p,q)# determine how far the centroid moves when dilated by a factor of #r#.

The centroid is #(u,v)=(1/3(a+c+e), 1/3(b+d+e))#

The distance #s# from the centroid to the dilation point is before dilation is

# s = sqrt{(u-p)^2 + (v-q)^2}#

After dilation it will be #rs# so the total distance moved is:

# m= (r-1) s = (r-1)sqrt{(u-p)^2 + (v-q)^2}#

#m = (r-1) sqrt { (1/3(a+c+e)-p)^2 + (1/3(b+d+e)-q)^2 }#

# m = {r-1}/3 sqrt{ (a+c+e-3p)^2 + (b+d+e-3q)^2 }#