# A triangle has corners at (6 ,9 ), (-2 ,-1 ), and (1 ,-1 ). If the triangle is dilated by a factor of 1/3  about point (-5 ,-2 ), how far will its centroid move?

Jul 14, 2018

color(blue)("Distance moved by centroid " color(crimson)(vec(GG') ~~ 5.3008 " units"

#### Explanation:

$A \left(6 , 9\right) , B \left(- 2 , - 1\right) , C \left(1 , - 1\right) , \text{ about point " D (-5, -2), " dilation factor } \frac{1}{3}$

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

$G \left(x , y\right) = \left(\frac{6 - 2 + 1}{3} , \frac{9 - 1 - 1}{3}\right) = \left(\frac{5}{3} , \frac{7}{3}\right)$

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = \frac{1}{3} a - - \frac{2}{3} d = \frac{1}{3} \cdot \left(\begin{matrix}6 \\ 9\end{matrix}\right) - - \frac{2}{3} \cdot \left(\begin{matrix}- 5 \\ - 2\end{matrix}\right) = \left(\begin{matrix}- \frac{4}{3} \\ \frac{5}{3}\end{matrix}\right)$

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

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

$\text{New Centroid } G ' \left(x , y\right) = \left(\frac{- \frac{4}{3} - 4 - 3}{3} , \frac{\frac{5}{3} - \frac{5}{3} - \frac{5}{3}}{3}\right) = \left(- \frac{25}{9} , - \frac{5}{9}\right)$

color(blue)("Distance moved by centroid " 

color(crimson)(vec(GG') = sqrt((5/3- -25/9)^2 + (7/3 - -5/9)) ~~ 5.3008 " units"#