# A triangle has corners at (2, -5), (-8, 4), and (1, -3). If the triangle is dilated by a factor of 2/5 about point (-1, -6), how far will its centroid move?

Jul 14, 2018

color(blue)(vec(GG') = sqrt((-5/3- -19/5)^2 + (-4/3- -76/15)) ~~ 3.7547 " units"

#### Explanation:

$A \left(2 , - 5\right) , B \left(- 8 , 4\right) , C \left(1 , - 3\right) , \text{ about point " D (-1,-6), " dilation factor } \frac{2}{5}$

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

$G \left(x , y\right) = \left(\frac{2 - 8 + 1}{3} , \frac{- 5 + 4 - 3}{3}\right) = \left(- \frac{5}{3} , - \frac{4}{3}\right)$

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\frac{2}{5}\right) a - \left(- \frac{3}{5}\right) d = \left(\frac{2}{5}\right) \cdot \left(\begin{matrix}2 \\ - 5\end{matrix}\right) + \left(\frac{3}{5}\right) \cdot \left(\begin{matrix}- 1 \\ - 6\end{matrix}\right) = \left(\begin{matrix}\frac{1}{5} \\ - \frac{33}{5}\end{matrix}\right)$

$B ' \left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\frac{2}{5}\right) b - \left(- \frac{3}{5}\right) d = \left(\frac{2}{5}\right) \cdot \left(\begin{matrix}- 8 \\ 4\end{matrix}\right) + \left(\frac{3}{5}\right) \cdot \left(\begin{matrix}- 1 \\ - 6\end{matrix}\right) = \left(\begin{matrix}- \frac{19}{5} \\ - 2\end{matrix}\right)$

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\frac{2}{5}\right) c - \left(- \frac{3}{5}\right) d = \left(\frac{2}{5}\right) \cdot \left(\begin{matrix}1 \\ - 3\end{matrix}\right) + \left(\frac{3}{5}\right) \cdot \left(\begin{matrix}- 1 \\ - 6\end{matrix}\right) = \left(\begin{matrix}- \frac{1}{5} \\ - \frac{23}{5}\end{matrix}\right)$

"New centroid " G'(x,y) = ((1/5 - 19/5 - 1/5)/3,(-33/5 - 2 - 23/5) /3 = (-19/15,-76/15)

color(blue)("Distance moved by centroid "

color(blue)(vec(GG') = sqrt((-5/3- -19/5)^2 + (-4/3- -76/15)) ~~ 3.7547 " units"