# A triangle has corners at (-1 ,2 ), (3 ,-5 ), and (7 ,-3 ). If the triangle is dilated by a factor of 5  about point (-2 ,4 ), how far will its centroid move?

Jul 14, 2018

color(purple)("Distance moved by centroid " 

color(blue)(vec(GG') ~~ 28.2843 " units"

#### Explanation:

$A \left(- 1 , 2\right) , B \left(3 , - 5\right) , C \left(7 , - 3\right) , \text{ about point " D (-2, 4), " dilation factor } 5$

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

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

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 5 a - 4 d = 5 \cdot \left(\begin{matrix}- 1 \\ 2\end{matrix}\right) - 4 \cdot \left(\begin{matrix}- 2 \\ 4\end{matrix}\right) = \left(\begin{matrix}3 \\ 18\end{matrix}\right)$

B'((x),(y)) = 5b - 4d = 5*((3),(-5)) - 4*(-2),(4)) = ((23),(-41))

$C ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 5 c - 4 d = 5 \cdot \left(\begin{matrix}7 \\ - 3\end{matrix}\right) - 4 \cdot \left(\begin{matrix}- 2 \\ 4\end{matrix}\right) = \left(\begin{matrix}43 \\ - 31\end{matrix}\right)$

$\text{New Centroid } G ' \left(x , y\right) = \left(\frac{3 + 23 + 43}{3} , \frac{18 - 41 - 31}{3}\right) = \left(23 , - 18\right)$

color(purple)("Distance moved by centroid " 

color(blue)(vec(GG') = sqrt((3-23)^2 + (-2- -18)) ~~ 28.2843 " units"#