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

Jul 22, 2018

color(green)("Distance moved by centroid " 

color(blue)(vec(GG') ~~ 14.3604 " units"

#### Explanation:

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

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

$G \left(x , y\right) = \left(\frac{1 + 2 - 8}{3} , \frac{3 - 2 + 6}{3}\right) = \left(- \frac{5}{3} , - \frac{7}{3}\right)$

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 5 a - 4 d = 5 \cdot \left(\begin{matrix}1 \\ 3\end{matrix}\right) - 4 \cdot \left(\begin{matrix}- 3 \\ 2\end{matrix}\right) = \left(\begin{matrix}17 \\ 7\end{matrix}\right)$

$B ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 5 b - 4 d = 5 \cdot \left(\begin{matrix}2 \\ - 2\end{matrix}\right) - 4 \cdot \left(\begin{matrix}- 3 \\ 2\end{matrix}\right) = \left(\begin{matrix}22 \\ - 18\end{matrix}\right)$

$C ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 5 c - 4 d = 5 \cdot \left(\begin{matrix}- 8 \\ 6\end{matrix}\right) - 4 \cdot \left(\begin{matrix}- 3 \\ 2\end{matrix}\right) = \left(\begin{matrix}- 28 \\ - 22\end{matrix}\right)$

$\text{New Centroid } G ' \left(x , y\right) = \left(\frac{17 + 22 - 28}{3} , \frac{7 - 18 - 22}{3}\right) = \left(\frac{11}{3} , - 11\right)$

color(green)("Distance moved by centroid " 

color(blue)(vec(GG') = sqrt((-5/3 - 11/3)^2 + (-7/3 - 11)) ~~ 14.3604 " units"#