# A triangle has corners at (8 ,3 ), (4 ,-5 ), and (-7 ,-4 ). If the triangle is dilated by a factor of 5  about point (1 ,-3 ), how far will its centroid move?

Feb 10, 2018

Distance moved by centroid after dilation color(green)(vec(GG') ~~ 11.39

#### Explanation:

Given : Points $A \left(\begin{matrix}8 \\ 3\end{matrix}\right) , B \left(\begin{matrix}4 \\ - 5\end{matrix}\right) , C \left(\begin{matrix}- 7 \\ - 4\end{matrix}\right)$, $D \left(\begin{matrix}1 \\ - 3\end{matrix}\right)$

Dilated by factor 5 about point $D \left(\begin{matrix}1 \\ - 3\end{matrix}\right)$

To find the distance moved by G, the centroid

Present coordinates of centroid G

G((x),(y)) -> (((8+4-7)/3), ((3-5-4)/3)) =>color(purple)( ((-5/3),(-2))

Let $G ' \left(\begin{matrix}x \\ y\end{matrix}\right)$ be the new coordinates of G

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

$\left(\begin{matrix}x - 1 \\ y + 3\end{matrix}\right) = 5 \cdot \left(\begin{matrix}\left(- \frac{5}{3}\right) - 1 \\ \left(- 2\right) + 3\end{matrix}\right)$

$x - 1 = 5 \left(- \frac{5}{3} - 1\right)$

$3 x - 3 = - 25 - 15 , 3 x = - 37 , x = - \frac{37}{3}$

$y + 3 = 5 \left(- 2 + 3\right) , y = 2$

G' color(brown)(((-37/3),(2))

vec(G'G) = sqrt((-5/3+37/3)^2 + (-2 -2)^2) ~~ color(green)(11.39#