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

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Answer 1 · 2017-03-30T06:15:33

The centroid will move by $= 8.69$ $=8.69$

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

$A = (2, 2)$ $A=(2,2)$

$B = (4, - 7)$ $B=(4,-7)$

$C = (3, 4)$ $C=(3,4)$

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

$C_{c} = (\frac{2 + 4 + 3}{3}, \frac{2 + (- 7) + 4}{3}) = (3, - \frac{1}{3})$ $C_c=((2+4+3)/3,(2+(-7)+4)/3)=(3,-1/3)$

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

The center of dilatation is $D=(1,-9)$

$vec(DA')=5vec(DA)=5*<1,11> = <5,55>$

$A'=(5+1,55-9)=(6,46)$

$vec(DB')=5vec(DB)=5*<3,2> = <15,10>$

$B'=(15+1,10-9)=(16,1)$

$vec(DC')=5vec(DC)=5*<2,13> = <10,65>$

$C'=(10+1,65-9)=(11,56)$

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

$C_c'=((6+16+11)/3,(46+1+56)/3)=(11,103/3)$

The distance between the 2 centroids is

$C_cC_c'=sqrt((11-3)^2+(103/3+1/3)^2)$

$=1/3sqrt(64*9+104^2)=26.08/3=8.69$

A triangle has corners at (2 ,2 )(2,2)(2 ,2 ), (4 ,-7 )(4,−7)(4 ,-7 ), and (3 ,4 )(3,4)(3 ,4 ). If the triangle is dilated by a factor of 5 55 about point #(1 ,-9 ), how far will its centroid move?