# A triangle has corners at (6 ,3 ), (4 ,-1 ), and (3 ,-9 ). If the triangle is dilated by a factor of 5  about point (8 ,-6 ), how far will its centroid move?

Jun 8, 2018

$\text{Distance moved by centroid is " = color(blue)(vec(GG') = 16.67 " units}$

#### Explanation:

$A \left(6 , 3\right) , B \left(4 , - 1\right) , C \left(3 , - 9\right) , \text{ about " D (8,-6), " with dilating factor } 5$

let G be the centroid.

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

$G \left(x , y\right) = G \left(\frac{6 + 4 + 3}{3} , \frac{3 - 1 - 9}{3}\right) = G \left(\frac{13}{3} , - \frac{7}{3}\right)$

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 5 a - 4 d = 5 \cdot \left(\begin{matrix}6 \\ 3\end{matrix}\right) - 4 \left(\begin{matrix}8 \\ - 6\end{matrix}\right) = \left(\begin{matrix}- 2 \\ 39\end{matrix}\right)$

$B ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 5 b - 4 d = 5 \cdot \left(\begin{matrix}4 \\ - 1\end{matrix}\right) - 4 \left(\begin{matrix}8 \\ - 6\end{matrix}\right) = \left(\begin{matrix}- 12 \\ 19\end{matrix}\right)$

$C ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 5 c - 4 d = 5 \cdot \left(\begin{matrix}3 \\ - 9\end{matrix}\right) - 4 \left(\begin{matrix}8 \\ - 6\end{matrix}\right) = \left(\begin{matrix}- 17 \\ - 21\end{matrix}\right)$

$G ' \left(x , y\right) = G ' \left(\frac{- 2 - 12 - 17}{3} , \frac{39 + 19 - 21}{3}\right) = G ' \left(- 9 , \frac{37}{3}\right)$

Distance moved by centroid is $\vec{G G '}$

color(blue)(vec(GG') = sqrt((-9-13/3)^2 + (37/3-7/3)^2) = 16.67, " units"#