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

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Answer 1 · 2017-04-10T10:02:51

The centroid will move by $= 18.52$ $=18.52$

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

$A = (- 2, 1)$ $A=(-2,1)$

$B = (8, - 5)$ $B=(8,-5)$

$C = (- 1, - 2)$ $C=(-1,-2)$

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

$C_{c} = (\frac{- 2 + 8 - 1}{3}, \frac{1 - 5 - 2}{3}) = (\frac{5}{3}, - 2)$ $C_c=((-2+8-1)/3,(1-5-2)/3)=(5/3,-2)$

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

The center of dilatation is $D=(4,-6)$

$vec(DA')=5vec(DA)=5*<-6,7> = <-30,35>$

$A'=(-30+4,35-6)=(-26,29)$

$vec(DB')=5vec(DB)=5*<4,1> = <20,5>$

$B'=(20+4,5-6)=(24,-1)$

$vec(DC')=5vec(DC)=5*<-5,4> = <-25,20>$

$C'=(-25+4,20-6)=(-21,14)$

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

$C_c'=((-26+24-21)/3,(29-1+14)/3)=(-23/3,42/3)$

The distance between the 2 centroids is

$C_cC_c'=sqrt((-23/3-5/3)^2+(42/3+2)^2)$

$=1/3sqrt(28^2+48^2)=55.57/3=18.52$

A triangle has corners at (-2 ,1 )(−2,1)(-2 ,1 ), (8 ,-5 )(8,−5)(8 ,-5 ), and (-1 ,-2 )(−1,−2)(-1 ,-2 ). If the triangle is dilated by a factor of 5 55 about point #(4 ,-6 ), how far will its centroid move?