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

Aug 8, 2018

color(purple)("Distance moved by centroid " color(chocolate)(vec(GG') = = sqrt 10 ~~ 3.1623 " units"

Explanation:

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

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

$G \left(x , y\right) = \left(\frac{4 + 5 + 2}{3} , \frac{5 + 3 + 1}{3}\right) = \left(\frac{11}{3} , 3\right)$

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 4 a - 3 d = 4 \cdot \left(\begin{matrix}4 \\ 5\end{matrix}\right) - 3 \left(\begin{matrix}4 \\ 2\end{matrix}\right) = \left(\begin{matrix}4 \\ 14\end{matrix}\right)$

B'((x),(y)) = 4 b - 3 d = 4 *((5),(3)) - 3 ((4),(2))) = ((8),(6))

$C ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 4 c - 3 d = 4 \cdot \left(\begin{matrix}2 \\ 1\end{matrix}\right) - 3 \left(\begin{matrix}4 \\ 2\end{matrix}\right) = \left(\begin{matrix}- 4 \\ - 2\end{matrix}\right)$

$\text{New Centroid } G ' \left(x , y\right) = \left(\frac{4 + 8 - 4}{3} , \frac{14 + 6 - 2}{3}\right) = \left(\frac{8}{3} , 6\right)$

color(purple)("Distance moved by centroid " 

color(chocolate)(vec(GG') = sqrt((11/3- 8/3)^2 + (3 - 6)^2) = sqrt 10 ~~ 3.1623 " units"#