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

Jul 14, 2018

color(green)("Distance moved by centroid " color(crimson)(vec(GG') ~~ 4.4721 " units"

#### Explanation:

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

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

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

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

B'((x),(y)) = 1/3b - -2/3d = 1/3*((5),(6)) - -2/3*(2),(-1)) = ((3),4/3))

$C ' \left(\begin{matrix}x \\ y\end{matrix}\right) = \frac{1}{3} c - - \frac{2}{3} d = \frac{1}{3} \cdot \left(\begin{matrix}- 4 \\ 7\end{matrix}\right) - - \frac{2}{3} \cdot \left(\begin{matrix}2 \\ - 1\end{matrix}\right) = \left(\begin{matrix}0 \\ \frac{5}{3}\end{matrix}\right)$

$\text{New Centroid } G ' \left(x , y\right) = \left(\frac{\frac{1}{3} + 3 + 0}{3} , \frac{- 1 + \frac{4}{3} + \frac{5}{3}}{3}\right) = \left(\frac{10}{3} , 2\right)$

color(green)("Distance moved by centroid " 

color(crimson)(vec(GG') = sqrt((-2/3- 10/3)^2 + (4 - 2)) ~~ 4.4721 " units"#