# A triangle has corners at (4 ,-2 ), (-1 ,2 ), and (1 ,3 ). If the triangle is dilated by a factor of 1/3  about point (6 ,8 ), how far will its centroid move?

Jun 19, 2018

color(crimson)(vec(GG') = sqrt((-32/9-4/3)^2 + (-1-1)) = 14.8 " units"

#### Explanation:

$A \left(- 2 , 6\right) , B \left(4 , - 3\right) , C \left(2 , 9\right) , \text{ about point " D (6,8), " dilation factor } \frac{1}{3}$

Centroid $G \left(x , y\right) = \left(\frac{{x}_{a} + {x}_{b} + {x}_{c}}{3} , \frac{{y}_{a} + {y}_{b} + {y}_{c}}{3}\right)$

$G \left(x , y\right) = \left(\frac{4 - 1 + 1}{3} , \frac{- 2 + 2 + 3}{3}\right) = \left(\frac{4}{3} , 1\right)$

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = \frac{1}{3} a - \left(\frac{2}{3}\right) d = \left(\frac{1}{3}\right) \cdot \left(\begin{matrix}4 \\ - 2\end{matrix}\right) - \left(\frac{2}{3}\right) \cdot \left(\begin{matrix}6 \\ 8\end{matrix}\right) = \left(\begin{matrix}- \frac{8}{3} \\ 6\end{matrix}\right)$

$B ' \left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\frac{1}{3}\right) b - \left(\frac{2}{3}\right) d = \left(\frac{1}{3}\right) \cdot \left(\begin{matrix}- 1 \\ 2\end{matrix}\right) - \left(\frac{2}{3}\right) \cdot \left(\begin{matrix}6 \\ 8\end{matrix}\right) = \left(\begin{matrix}- \frac{13}{3} \\ - \frac{14}{3}\end{matrix}\right)$

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\frac{1}{3}\right) c - \left(\frac{2}{3}\right) d = \left(\frac{1}{3}\right) \cdot \left(\begin{matrix}1 \\ 3\end{matrix}\right) - \left(\frac{2}{3}\right) \cdot \left(\begin{matrix}6 \\ 8\end{matrix}\right) = \left(\begin{matrix}- \frac{11}{3} \\ - \frac{13}{3}\end{matrix}\right)$

$\text{New Centroid } G ' \left(x , y\right) = \left(\frac{- \frac{8}{3} - \frac{13}{3} - \frac{11}{3}}{3} , \frac{6 - \frac{14}{3} - \frac{13}{3}}{3}\right) = \left(- \frac{32}{9} , - 1\right)$

color(purple)("Distance moved by centroid " 

color(crimson)(vec(GG') = sqrt((-32/9-4/3)^2 + (-1-1)) = 14.8 " units"#