# A triangle as corners at (2 ,5 ), (7 ,2), and (3 ,1). If the triangle is dilated by a factor of 4  about (1 ,9), how far will its centroid move?

Feb 10, 2018

Distance moved by centroid due to dilation factor 4 is

$\textcolor{g r e e n}{G G ' \approx 16.36}$

#### Explanation:

Centroid $G \left(\begin{matrix}x \\ y\end{matrix}\right) \to \left(\begin{matrix}\frac{2 + 7 + 3}{3} \\ \frac{5 + 2 + 1}{3}\end{matrix}\right) \implies \left(\begin{matrix}4 \\ \frac{8}{3}\end{matrix}\right)$

color(blue)(G ((4),(8/3))

Dilated about point D ((1),(9)) by a factor of 4

$\vec{G ' D} = 4 \cdot \vec{G D}$

$\left(\begin{matrix}x - 1 \\ y - 9\end{matrix}\right) = 4 \cdot \left(\begin{matrix}4 - 1 \\ \frac{8}{3} - 9\end{matrix}\right) \implies 4 \left(\begin{matrix}3 \\ - \frac{19}{3}\end{matrix}\right)$

$x - 1 = 4 \cdot 3 = 12 , x = 13$

$y - 9 = 4 \cdot \left(- \frac{19}{3}\right) = - \frac{76}{3} , y = \frac{49}{3}$

color(blue)(G'((13),(49/3))

Distance moved by G to G'

$\vec{G G '} = \sqrt{{\left(13 - 4\right)}^{2} + {\left(\frac{49}{3} - \frac{8}{3}\right)}^{2}} \approx \textcolor{b r o w n}{16.36}$