A triangle has corners at (4 ,-6 ), (3 ,-3 ), and (8 ,4 ). If the triangle is dilated by a factor of 1/3  about point (-3 ,2 ), how far will its centroid move?

Feb 10, 2018

Distance moved by centroid ~~ color(green)(2.53

Explanation:

Formila for finding coordinates of centroid

$G \left(\begin{matrix}\frac{{x}_{1} + {x}_{2} + {x}_{3}}{c} \\ \frac{{y}_{1} + {y}_{2} + {y}_{3}}{3}\end{matrix}\right) = \left(\begin{matrix}\frac{4 + 3 + 8}{3} \\ \frac{- 6 - 3 + 4}{2}\end{matrix}\right) = \left(\begin{matrix}5 \\ - \frac{5}{3}\end{matrix}\right)$

Dilated around point $\left(\begin{matrix}- 3 \\ 2\end{matrix}\right)$ by a factor of (1/3)

New coordinates of centroid $\vec{G ' D} \left(\begin{matrix}x + 3 \\ y - 2\end{matrix}\right) = \left(\frac{1}{3}\right) \vec{G D} = \left(\frac{1}{3}\right) \left(\begin{matrix}5 + 3 \\ - \frac{5}{3} - 2\end{matrix}\right) \implies \left(\begin{matrix}\frac{8}{3} \\ - \frac{11}{9}\end{matrix}\right)$

$G ' \left(\begin{matrix}\frac{17}{3} \\ \frac{7}{9}\end{matrix}\right)$

Distance between G & G'

vec(GG') = sqrt((17/3-5)^2 + (7/9 + 5/3)^2) = sqrt((7/5)^2 + (4/9)^2) = ~~ color(green)(2.53