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

Jul 14, 2018

color(green)("Distance moved by centroid " color(purple)(vec(GG') ~~ 13.4164 " units"

#### Explanation:

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

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

$G \left(x , y\right) = \left(\frac{6 + 4 + 2}{3} , \frac{3 - 1 - 9}{3}\right) = \left(4 , - \frac{7}{3}\right)$

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 5 a - 4 d = 5 \cdot \left(\begin{matrix}6 \\ 3\end{matrix}\right) - 4 \cdot \left(\begin{matrix}1 \\ - 5\end{matrix}\right) = \left(\begin{matrix}26 \\ 35\end{matrix}\right)$

B'((x),(y)) = 5b - 4d = 5*((4),(-1)) - 4*(1),(-5)) = ((16),15))

$C ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 5 c - 4 d = 5 \cdot \left(\begin{matrix}2 \\ - 9\end{matrix}\right) - 4 \cdot \left(\begin{matrix}1 \\ - 5\end{matrix}\right) = \left(\begin{matrix}6 \\ - 25\end{matrix}\right)$

$\text{New Centroid } G ' \left(x , y\right) = \left(\frac{26 + 16 + 6}{3} , \frac{35 + 15 - 25}{3}\right) = \left(16 , \frac{25}{3}\right)$

color(green)("Distance moved by centroid " 

color(purple)(vec(GG') = sqrt((4- 16)^2 + (-7/3 + 25/3)) ~~ 13.4164 " units"#