# A triangle has corners at (6 ,2 ), (5 ,-8 ), and (-5 ,3 ). If the triangle is dilated by a factor of 5  about point (7 ,-2 ), how far will its centroid move?

Jul 14, 2018

color(indigo)(vec(GG') = sqrt((2- -18)^2 + (-1-3)) ~~ 20.396 " units"

#### Explanation:

$A \left(6 , 2\right) , B \left(5 , - 8\right) , C \left(- 5 , 3\right) , \text{ about point " D (7,-2), " dilation factor } 5$

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

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

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 5 a - d = 5 \cdot \left(\begin{matrix}6 \\ 2\end{matrix}\right) - 4 \cdot \left(\begin{matrix}7 \\ - 2\end{matrix}\right) = \left(\begin{matrix}2 \\ 18\end{matrix}\right)$

$B ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 5 b - d = 5 \cdot \left(\begin{matrix}5 \\ - 8\end{matrix}\right) - 4 \cdot \left(\begin{matrix}7 \\ - 2\end{matrix}\right) = \left(\begin{matrix}- 3 \\ - 32\end{matrix}\right)$

$C ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 5 c - d = 5 \cdot \left(\begin{matrix}- 5 \\ 3\end{matrix}\right) - 4 \cdot \left(\begin{matrix}7 \\ - 2\end{matrix}\right) = \left(\begin{matrix}- 53 \\ 23\end{matrix}\right)$

$\text{New Centroid } G ' \left(x , y\right) = \left(\frac{2 - 3 - 53}{3} , \frac{18 - 32 + 23}{3}\right) = \left(- 18 , 3\right)$

color(indigo)("Distance moved by centroid " 

color(indigo)(vec(GG') = sqrt((2- -18)^2 + (-1-3)) ~~ 20.396 " units"#