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

Apr 10, 2017

The centroid will move by $= 18.52$

#### Explanation:

Let $A B C$ be the triangle

$A = \left(- 2 , 1\right)$

$B = \left(8 , - 5\right)$

$C = \left(- 1 , - 2\right)$

The centroid of triangle $A B C$ is

${C}_{c} = \left(\frac{- 2 + 8 - 1}{3} , \frac{1 - 5 - 2}{3}\right) = \left(\frac{5}{3} , - 2\right)$

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

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

$\vec{D A '} = 5 \vec{D A} = 5 \cdot < - 6 , 7 > = < - 30 , 35 >$

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

$\vec{D B '} = 5 \vec{D B} = 5 \cdot < 4 , 1 > = < 20 , 5 >$

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

$\vec{D C '} = 5 \vec{D C} = 5 \cdot < - 5 , 4 > = < - 25 , 20 >$

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

The centroid ${C}_{c} '$ of triangle $A ' B ' C '$ is

${C}_{c} ' = \left(\frac{- 26 + 24 - 21}{3} , \frac{29 - 1 + 14}{3}\right) = \left(- \frac{23}{3} , \frac{42}{3}\right)$

The distance between the 2 centroids is

${C}_{c} {C}_{c} ' = \sqrt{{\left(- \frac{23}{3} - \frac{5}{3}\right)}^{2} + {\left(\frac{42}{3} + 2\right)}^{2}}$

$= \frac{1}{3} \sqrt{{28}^{2} + {48}^{2}} = \frac{55.57}{3} = 18.52$