# A triangle has corners at (-2 ,6 ), (4 ,-3 ), and (2 ,9 ). If the triangle is dilated by a factor of 5  about point (-7 ,1 ), how far will its centroid move?

Jun 8, 2018

color(purple)("Distance moved by centroid " 

color(crimson)(vec(GG') = 35.9 " units"

#### Explanation:

$A \left(- 2 , 6\right) , B \left(4 , - 3\right) , C \left(2 , 9\right) , \text{ about point " D (-7,10), " dilation factor } 5$

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

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

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 5 a - 4 d = 5 \cdot \left(\begin{matrix}- 2 \\ 6\end{matrix}\right) - 4 \cdot \left(\begin{matrix}- 7 \\ 1\end{matrix}\right) = \left(\begin{matrix}18 \\ 26\end{matrix}\right)$

$B ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 5 b - 4 d = 5 \cdot \left(\begin{matrix}4 \\ - 3\end{matrix}\right) - 4 \cdot \left(\begin{matrix}- 7 \\ 1\end{matrix}\right) = \left(\begin{matrix}48 \\ - 19\end{matrix}\right)$

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 5 c - 4 d = 5 \cdot \left(\begin{matrix}2 \\ 9\end{matrix}\right) - 4 \cdot \left(\begin{matrix}- 7 \\ 1\end{matrix}\right) = \left(\begin{matrix}38 \\ 45\end{matrix}\right)$

$\text{New Centroid } G ' \left(x , y\right) = \left(\frac{18 + 48 + 38}{3} , \frac{26 - 19 + 45}{3}\right) = \left(\frac{104}{3} , \frac{52}{3}\right)$

color(purple)("Distance moved by centroid " 

color(crimson)(vec(GG') = sqrt((104/3-4/3)^2 + (52/3-4)) = 35.9 " units"#