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

Jun 7, 2018

color(crimson)("Centroid will move by " 8.25 " units"

#### Explanation:

$A \left(1 , 2\right) , B \left(- 3 , - 1\right) , C \left(4 , - 2\right)$, dilated around $D \left(- 5 , 7\right)$ by a factor of 2.

$\overline{A ' D} = 2 \cdot \overline{A D}$

$A ' - d = 2 \cdot \left(a - d\right)$

$A ' = 2 a - d$

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

Similarly,

$B ' = 2 b - d$

$B ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 2 \left(\begin{matrix}- 3 \\ 1\end{matrix}\right) - \left(\begin{matrix}- 5 \\ 7\end{matrix}\right) = \left(\begin{matrix}- 1 \\ - 5\end{matrix}\right)$

$C ' = 2 c - d$

$C ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 2 \left(\begin{matrix}4 \\ - 2\end{matrix}\right) - \left(\begin{matrix}- 5 \\ 7\end{matrix}\right) = \left(\begin{matrix}13 \\ - 11\end{matrix}\right)$

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

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

$\text{Centroid of Traingle A'B'C', } G ' = \left(\frac{{x}_{a} ' + {x}_{b} ' + {x}_{c} '}{3} , \frac{{y}_{a} ' + {y}_{b} ' + {y}_{c} '}{3}\right)$

$G ' \left(x , y\right) = \left(\frac{7 - 1 + 13}{3} , \frac{- 3 - 5 - 11}{3}\right) = \left(\frac{19}{3} , - \frac{19}{3}\right)$

vec(GG') ' gives the distance moved by the centroid"#

$\vec{G G '} = \sqrt{{\left(\frac{2}{3} - \frac{19}{3}\right)}^{2} + {\left(- \frac{1}{3} + \frac{19}{3}\right)}^{2}} = 8.25$