# A triangle has corners at (-7 ,-5 ), (-2 ,-1 ), and (3 ,7 ). If the triangle is dilated by a factor of 2/5  about point (1 ,-8 ), how far will its centroid move?

Jul 14, 2018

color(green)(vec(GG') = sqrt((-2- -1/5)^2 + (1/3- -14/3)) ~~ 5.314 " units"

#### Explanation:

$A \left(- 7 , - 5\right) , B \left(- 2 , - 1\right) , C \left(3 , 7\right) , \text{ about point " D (1,-8), " dilation factor } \frac{2}{5}$

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

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

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\frac{2}{5}\right) a - \left(- \frac{3}{5}\right) d = \left(\frac{2}{5}\right) \cdot \left(\begin{matrix}- 7 \\ - 5\end{matrix}\right) + \left(\frac{3}{5}\right) \cdot \left(\begin{matrix}1 \\ - 8\end{matrix}\right) = \left(\begin{matrix}- \frac{11}{5} \\ - \frac{34}{5}\end{matrix}\right)$

$B ' \left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\frac{2}{5}\right) b - \left(- \frac{3}{5}\right) d = \left(\frac{2}{5}\right) \cdot \left(\begin{matrix}- 2 \\ - 1\end{matrix}\right) + \left(\frac{3}{5}\right) \cdot \left(\begin{matrix}1 \\ - 8\end{matrix}\right) = \left(\begin{matrix}- \frac{1}{5} \\ - \frac{26}{5}\end{matrix}\right)$

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\frac{2}{5}\right) c - \left(- \frac{3}{5}\right) d = \left(\frac{2}{5}\right) \cdot \left(\begin{matrix}3 \\ 7\end{matrix}\right) + \left(\frac{3}{5}\right) \cdot \left(\begin{matrix}1 \\ - 8\end{matrix}\right) = \left(\begin{matrix}\frac{9}{5} \\ - 2\end{matrix}\right)$

"New centroid " G'(x,y) = ((-11/5 - 1/5 + 9/5)/3,(-34/5 - 26/5 - 2) /3 = (-1/5,-14/3)

color(green)("Distance moved by centroid " 

color(green)(vec(GG') = sqrt((-2- -1/5)^2 + (1/3- -14/3)) ~~ 5.314 " units"#