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

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Answer 1 · 2017-03-14T19:29:38

The distance is $= 5.83$ $=5.83$

Let $A B C$ $ABC$ be the triangle

$A = (5.3)$ $A=(5.3)$

$B = (1, 4)$ $B=(1,4)$

$C = (3.5)$ $C=(3.5)$

The centroid of triangle $A B C$ $ABC$ is

$C_{c} = (\frac{5 + 1 + 3}{3}, \frac{3 + 4 + 5}{3}) = (3, 4)$ $C_c=((5+1+3)/3,(3+4+5)/3)=(3,4)$

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

The center of dilatation is $D=(2,2)$

$vec(DA')=3vec(DA)=3*<3,1> = <9,3>$

$A'=(9+2,3+2)=(11,5)$

$vec(DB')=3vec(DB)=3*<-1,2> = <-3,6>$

$B'=(-3+2,6+2)=(-1,11)$

$vec(DC')=3vec(Dc)=3*<2,3> = <6,9>$

$C'=(6+2,9+2)=(8,11)$

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

$C_c'=((11-1+8)/3,(5+11+11)/3)=(6,9)$

The distance between the 2 centroids is

$C_cC_c'=sqrt((6-3)^2+(9-4)^2)$

$=sqrt(9+25)=sqrt34=5.83$

A triangle as corners at (5 ,3 )(5,3)(5 ,3 ), (1 ,4 )(1,4)(1 ,4 ), and (3 ,5 )(3,5)(3 ,5 ). If the triangle is dilated by a factor of 3 33 about #(2 ,2 ), how far will its centroid move?