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

Jul 14, 2018

color(indigo)("Distance moved by centroid " color(crimson)(vec(GG') ~~ 3.02255 " units"

#### Explanation:

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

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

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

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

$B ' \left(\begin{matrix}x \\ y\end{matrix}\right) = \frac{1}{3} b - - \frac{2}{3} d = \frac{1}{3} \cdot \left(\begin{matrix}7 \\ 3\end{matrix}\right) - - \frac{2}{3} \cdot \left(\begin{matrix}- 3 \\ 4\end{matrix}\right) = \left(\begin{matrix}\frac{1}{3} \\ \frac{11}{3}\end{matrix}\right)$

$C ' \left(\begin{matrix}x \\ y\end{matrix}\right) = \frac{1}{3} c - - \frac{2}{3} d = \frac{1}{3} \cdot \left(\begin{matrix}1 \\ 2\end{matrix}\right) - - \frac{2}{3} \cdot \left(\begin{matrix}- 3 \\ 4\end{matrix}\right) = \left(\begin{matrix}- \frac{5}{3} \\ \frac{10}{3}\end{matrix}\right)$

$\text{New Centroid } G ' \left(x , y\right) = \left(\frac{- 3 + \frac{1}{3} - \frac{5}{3}}{3} , \frac{\frac{7}{3} + \frac{11}{3} + \frac{10}{3}}{3}\right) = \left(- \frac{7}{9} , \frac{28}{9}\right)$

color(indigo)("Distance moved by centroid " 

color(crimson)(vec(GG') = sqrt((5/3- -7/9)^2 + (4/3 -28/9)) ~~ 3.02255 " units"#