# A triangle has corners at (2 ,1 ), (4 ,-3 ), and (-1 ,4 ). If the triangle is dilated by a factor of 5  about point #(4 ,-9 ), how far will its centroid move?

Feb 10, 2018

Centroid moves by $\approx \textcolor{g r e e n}{54.62}$

#### Explanation:

Centroid $G \left(\begin{matrix}\frac{2 + 4 - 1}{3} \\ \frac{1 - 3 + 4}{3}\end{matrix}\right) \implies \left(\begin{matrix}- \frac{5}{3} \\ \frac{2}{3}\end{matrix}\right)$

Dilated about D(4-9) by a factor of 5

To find new coordinate of centroid G'

$\vec{G ' D} = 5 \cdot \vec{G D}$

$G ' \left(\begin{matrix}x - 4 \\ y + 9\end{matrix}\right) = 5 \cdot \left(\begin{matrix}- \frac{5}{3} - 4 \\ \frac{2}{3} + 9\end{matrix}\right) \implies \left(\begin{matrix}- \frac{85}{3} \\ \frac{145}{3}\end{matrix}\right)$

$G ' \left(\begin{matrix}- \frac{73}{3} \\ \frac{118}{3}\end{matrix}\right)$

$\vec{G G '} = \sqrt{{\left(- \frac{5}{3} + \frac{85}{3}\right)}^{2} + {\left(\frac{2}{3} - \frac{145}{3}\right)}^{2}} \approx \textcolor{g r e e n}{54.62}$