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

Jul 14, 2018

color(maroon)("Distance moved by centroid " color(indigo)(vec(GG') ~~ 3.6667 " units"

Explanation:

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

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

$G \left(x , y\right) = \left(\frac{5 + 9 + 2}{3} , \frac{3 + 4 + 2}{3}\right) = \left(\frac{16}{3} , 3\right)$

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 2 a - d = 2 \cdot \left(\begin{matrix}5 \\ 3\end{matrix}\right) - 1 \cdot \left(\begin{matrix}9 \\ 1\end{matrix}\right) = \left(\begin{matrix}1 \\ 5\end{matrix}\right)$

B'((x),(y)) = 2b - d = 2*((9),(4)) - 1*(9),(1)) = ((9),(7))

$C ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 2 c - d = 2 \cdot \left(\begin{matrix}2 \\ 2\end{matrix}\right) - 1 \cdot \left(\begin{matrix}9 \\ 1\end{matrix}\right) = \left(\begin{matrix}- 5 \\ 3\end{matrix}\right)$

$\text{New Centroid } G ' \left(x , y\right) = \left(\frac{1 + 9 - 5}{3} , \frac{5 + 7 - 3}{3}\right) = \left(\frac{5}{3} , 3\right)$

color(maroon)("Distance moved by centroid " 

color(indigo)(vec(GG') = sqrt((16/3- 5/3)^2 + (3- 3)) ~~ 3.6667 " units"#