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

Jul 14, 2018

color(purple)(vec(GG') = sqrt((22/3-7)^2 + (5-3)) ~~ 2.0276 " units"

#### Explanation:

$A \left(7 , 3\right) , B \left(9 , 4\right) , C \left(5 , 2\right) , \text{ about point " D (6,1), " dilation factor } 2$

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

$G \left(x , y\right) = \left(\frac{7 + 9 + 5}{3} , \frac{3 + 4 + 2}{3}\right) = \left(7 , 3\right)$

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 2 a - d = 2 \cdot \left(\begin{matrix}7 \\ 3\end{matrix}\right) - \left(\begin{matrix}6 \\ 1\end{matrix}\right) = \left(\begin{matrix}8 \\ 5\end{matrix}\right)$

$B ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 2 b - d = 2 \cdot \left(\begin{matrix}8 \\ 4\end{matrix}\right) - \left(\begin{matrix}6 \\ 1\end{matrix}\right) = \left(\begin{matrix}10 \\ 7\end{matrix}\right)$

C'((x),(y)) = 2c - d = 2*((5),(2)) - ((6),(1) = ((4),(3))

$\text{New Centroid } G ' \left(x , y\right) = \left(\frac{8 + 10 + 4}{3} , \frac{5 + 7 + 3}{3}\right) = \left(\frac{22}{3} , 5\right)$

color(purple)("Distance moved by centroid " 

color(purple)(vec(GG') = sqrt((22/3-7)^2 + (5-3)) ~~ 2.0276 " units"#