# A triangle has corners at (-2 ,1 ), (6 ,-3 ), and (-1 ,4 ). If the triangle is dilated by a factor of 5  about point (4 ,-9 ), how far will its centroid move?

Jul 14, 2018

color(crimson)(vec(GG') = sqrt((-1/3-1)^2 + (118/3-2/3)) ~~ 38.69 " units"

#### Explanation:

$A \left(- 2 , 1\right) , B \left(6 , - 3\right) , C \left(- 1 , 4\right) , \text{ about point " D (4,-9), " dilation factor } 5$

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

$G \left(x , y\right) = \left(\frac{- 2 + 6 - 1}{3} , \frac{1 - 3 + 4}{3}\right) = \left(1 , \frac{2}{3}\right)$

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 5 a - 4 d = 5 \cdot \left(\begin{matrix}- 2 \\ 1\end{matrix}\right) - 4 \cdot \left(\begin{matrix}4 \\ - 9\end{matrix}\right) = \left(\begin{matrix}- 26 \\ 41\end{matrix}\right)$

$B ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 5 b - 4 d = 5 \cdot \left(\begin{matrix}6 \\ - 3\end{matrix}\right) - 4 \cdot \left(\begin{matrix}4 \\ - 9\end{matrix}\right) = \left(\begin{matrix}14 \\ 21\end{matrix}\right)$

$C ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 5 c - 4 d = 5 \cdot \left(\begin{matrix}- 1 \\ 4\end{matrix}\right) - 4 \cdot \left(\begin{matrix}4 \\ - 9\end{matrix}\right) = \left(\begin{matrix}11 \\ 56\end{matrix}\right)$

$\text{New Centroid } G ' \left(x , y\right) = \left(\frac{- 26 + 14 + 11}{3} , \frac{41 + 21 + 56}{3}\right) = \left(- \frac{1}{3} , \frac{118}{3}\right)$

color(purple)("Distance moved by centroid " 

color(crimson)(vec(GG') = sqrt((-1/3-1)^2 + (118/3-2/3)) ~~ 38.69 " units"#