# A triangle has corners at (2 ,-2 ), (3 ,-1 ), and (5 ,7 ). If the triangle is dilated by a factor of 2/5  about point (6 ,-2 ), how far will its centroid move?

Jul 14, 2018

color(indigo)("Distance moved by centroid " color(magenta)(vec(GG') ~~ 3.3333" units"

#### Explanation:

$A \left(2 , - 2\right) , B \left(3 , - 1\right) , C \left(5 , 7\right) , \text{ about point " D (6,-2), " dilation factor } \frac{2}{5}$

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

$G \left(x , y\right) = \left(\frac{2 + 3 + 5}{3} , \frac{- 2 - 1 + 7}{3}\right) = \left(\frac{10}{3} , \frac{4}{3}\right)$

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = \frac{2}{5} a - - \frac{3}{5} d = \frac{2}{5} \cdot \left(\begin{matrix}2 \\ - 2\end{matrix}\right) - - \frac{3}{5} \cdot \left(\begin{matrix}6 \\ - 2\end{matrix}\right) = \left(\begin{matrix}\frac{22}{5} \\ - 2\end{matrix}\right)$

$B ' \left(\begin{matrix}x \\ y\end{matrix}\right) = \frac{2}{5} b - - \frac{3}{5} d = \frac{2}{5} \cdot \left(\begin{matrix}3 \\ - 1\end{matrix}\right) - - \frac{3}{5} \cdot \left(\begin{matrix}6 \\ - 2\end{matrix}\right) = \left(\begin{matrix}\frac{24}{5} \\ - \frac{8}{5}\end{matrix}\right)$

$C ' \left(\begin{matrix}x \\ y\end{matrix}\right) = \frac{2}{5} c - - \frac{3}{5} d = \frac{2}{5} \cdot \left(\begin{matrix}5 \\ 7\end{matrix}\right) - - \frac{3}{5} \cdot \left(\begin{matrix}6 \\ - 2\end{matrix}\right) = \left(\begin{matrix}\frac{28}{5} \\ \frac{8}{5}\end{matrix}\right)$

$\text{New Centroid } G ' \left(x , y\right) = \left(\frac{\frac{22}{5} + \frac{24}{5} + \frac{28}{5}}{3} , \frac{- 2 - \frac{8}{5} + \frac{8}{5}}{3}\right) = \left(\frac{74}{15} , - \frac{2}{3}\right)$

color(indigo)("Distance moved by centroid " 

color(magenta)(vec(GG') = sqrt((10/3- 74/15)^2 + (4/3 - -2/3)) ~~ 3.3333" units"#