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

Jun 8, 2018

color(crimson)("Centroid will by 25.75 units"#

#### Explanation:

$A \left(- 2 , 1\right) , b \left(8 , - 5\right) , C \left(- 1 , 4\right) , D \left(4 , - 6\right) \text{dilation factor } 5$

Centroid $G = \frac{{x}_{1} + {x}_{2} + {x}_{3}}{3} , \frac{{y}_{1} + {y}_{2} + {y}_{3}}{3}$

$G \left(x , y\right) = \left(\frac{- 2 + 8 - 1}{3} , \frac{1 - 5 + 4}{3}\right) = \left(\frac{5}{3} , 0\right)$

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 5 a - 4 d = 5 \cdot \left(\begin{matrix}- 2 \\ 1\end{matrix}\right) - 4 \cdot \left(\begin{matrix}4 \\ - 6\end{matrix}\right) = \left(\begin{matrix}- 26 \\ 29\end{matrix}\right)$

$B ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 5 b - 4 d = 5 \cdot \left(\begin{matrix}8 \\ - 5\end{matrix}\right) - 4 \cdot \left(\begin{matrix}4 \\ - 6\end{matrix}\right) = \left(\begin{matrix}24 \\ - 1\end{matrix}\right)$

$C ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 5 c - 4 d = 5 \cdot \left(\begin{matrix}- 1 \\ 4\end{matrix}\right) - 4 \cdot \left(\begin{matrix}4 \\ - 6\end{matrix}\right) = \left(\begin{matrix}- 21 \\ 44\end{matrix}\right)$

$G ' \left(x , y\right) = \left(\frac{- 26 + 24 - 21}{3} , \frac{29 - 1 + 44}{3}\right) = \left(- \frac{23}{3} , 24\right)$

$\overline{G G '} = \sqrt{{\left(- \frac{23}{3} - \frac{5}{3}\right)}^{2} + {24}^{2}} = 25.75$ units