# A triangle as corners at (2 ,7 ), (5 ,8 ), and (3 ,1 ). If the triangle is dilated by a factor of 3  about (6 ,5 ), how far will its centroid move?

##### 1 Answer
Jul 14, 2018

color(maroon)("Distance moved by centroid " color(green)(vec(GG') ~~ 1.4907 " units"

#### Explanation:

$A \left(2 , 7\right) , B \left(5 , 8\right) , C \left(3 , 1\right) , \text{ about point " D (6,5), " dilation factor } 3$

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

$G \left(x , y\right) = \left(\frac{2 + 5 + 3}{3} , \frac{7 + 8 + 1}{3}\right) = \left(\frac{10}{3} , - \frac{16}{3}\right)$

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 3 a - 2 d = 3 \cdot \left(\begin{matrix}2 \\ 7\end{matrix}\right) - 2 \cdot \left(\begin{matrix}6 \\ 5\end{matrix}\right) = \left(\begin{matrix}- 6 \\ 11\end{matrix}\right)$

B'((x),(y)) = 3b - 2d = 3*((5),(8)) - 2*(6),(5)) = ((3),(14))

$C ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 3 c - 2 d = 3 \cdot \left(\begin{matrix}3 \\ 1\end{matrix}\right) - 2 \cdot \left(\begin{matrix}6 \\ 5\end{matrix}\right) = \left(\begin{matrix}- 3 \\ - 7\end{matrix}\right)$

$\text{New Centroid } G ' \left(x , y\right) = \left(\frac{- 6 + 3 - 3}{3} , \frac{11 + 14 - 7}{3}\right) = \left(- 2 , 6\right)$

color(maroon)("Distance moved by centroid " 

color(green)(vec(GG') = sqrt((10/3- -2)^2 + (-16/3- 6)) ~~ 1.4907 " units"#