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

Jul 22, 2018

color(purple)("Distance moved by centroid " color(crimson)(vec(GG') ~~ 0.4969 " units"

#### Explanation:

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

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

$G \left(x , y\right) = \left(\frac{6 - 3 + 1}{3} , \frac{2 + 4 - 1}{3}\right) = \left(\frac{4}{3} , \frac{5}{3}\right)$

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = \frac{1}{3} a - \left(\frac{2}{3}\right) d = \left(\frac{1}{3}\right) \cdot \left(\begin{matrix}6 \\ 2\end{matrix}\right) - \left(\frac{2}{3}\right) \cdot \left(\begin{matrix}- 2 \\ - 2\end{matrix}\right) = \left(\begin{matrix}\frac{10}{3} \\ 2\end{matrix}\right)$

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

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\frac{1}{3}\right) c - \left(\frac{2}{3}\right) d = \left(\frac{1}{3}\right) \cdot \left(\begin{matrix}1 \\ - 1\end{matrix}\right) - \left(\frac{2}{3}\right) \cdot \left(\begin{matrix}- 2 \\ - 2\end{matrix}\right) = \left(\begin{matrix}\frac{5}{3} \\ 1\end{matrix}\right)$

$\text{New Centroid } G ' \left(x , y\right) = \left(\frac{\frac{10}{3} + \frac{1}{3} + \frac{5}{3}}{3} , \frac{2 + \frac{8}{3} + 1}{3}\right) = \left(\frac{16}{9} , \frac{17}{9}\right)$

color(purple)("Distance moved by centroid " 

color(crimson)(vec(GG') = sqrt((16/9-4/3)^2 + (17/9-5/3)) ~~ 0.4969 " units"#