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

Jul 14, 2018

color(crimson)(vec(GG') = sqrt((4/3-14/3)^2 + (7/3- -23/3)) ~~ 10.5409 " units"

#### Explanation:

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

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

$G \left(x , y\right) = \left(\frac{- 2 + 1 + 5}{3} , \frac{3 - 4 - 6}{3}\right) = \left(\frac{4}{3} , \frac{7}{3}\right)$

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

B'((x),(y)) = 2b - 1d = 2*((1),(-4)) - 1*(-2),(3)) = ((4),(-11))

$C ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 2 c - 1 d = 2 \cdot \left(\begin{matrix}5 \\ - 6\end{matrix}\right) - 1 \cdot \left(\begin{matrix}- 2 \\ 3\end{matrix}\right) = \left(\begin{matrix}12 \\ - 15\end{matrix}\right)$

$\text{New Centroid } G ' \left(x , y\right) = \left(\frac{- 2 + 4 + 12}{3} , \frac{3 - 11 - 15}{3}\right) = \left(\frac{14}{3} , - \frac{23}{3}\right)$

color(purple)("Distance moved by centroid " 

color(crimson)(vec(GG') = sqrt((4/3-14/3)^2 + (7/3- -23/3)) ~~ 10.5409 " units"#