# A triangle has corners at (8 ,2 ), (4 ,9 ), and (-5 ,3 ). If the triangle is dilated by a factor of 5  about point (1 ,-2 ), how far will its centroid move?

Feb 12, 2018

Distance moved by centroid is $\vec{G G '} \approx \textcolor{g r e e n}{28.83}$

#### Explanation:

Given : A (8,2), B (4,9), C(-5,3)

Dilated about D(1,-2) and dilation factor 5

$C e n t r o i d$G = (8+4+(-5))/3, (2+4+3)/3 = color(brown)((7/3,3)

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

$a ' - d = 5 \left(a - d\right)$ or $a ' = 5 a - 4 d$

$\implies 5 \left(\begin{matrix}8 \\ 2\end{matrix}\right) - 4 \left(\begin{matrix}1 \\ - 2\end{matrix}\right) = \left(\begin{matrix}40 \\ 10\end{matrix}\right) - \left(\begin{matrix}4 \\ - 8\end{matrix}\right) = \left(\begin{matrix}36 \\ 18\end{matrix}\right)$

color(blue)(A' (36, 18)

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

$b ' - d = 5 \left(b - d\right)$ or $b ' = 5 b - 4 d$

$\implies 5 \left(\begin{matrix}4 \\ 9\end{matrix}\right) - 4 \left(\begin{matrix}1 \\ - 2\end{matrix}\right) = \left(\begin{matrix}20 \\ 45\end{matrix}\right) - \left(\begin{matrix}4 \\ - 8\end{matrix}\right) = \left(\begin{matrix}16 \\ 53\end{matrix}\right)$

color(blue)(B' (16, 53)

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

$c ' - D = 5 \left(c - d\right)$ or $c ' = 5 c - 4 d$

$\implies 5 \left(\begin{matrix}- 5 \\ 3\end{matrix}\right) - 4 \left(\begin{matrix}1 \\ - 2\end{matrix}\right) = \left(\begin{matrix}- 25 \\ 15\end{matrix}\right) - \left(\begin{matrix}4 \\ - 8\end{matrix}\right) = \left(\begin{matrix}- 29 \\ 23\end{matrix}\right)$

color(blue)(C' (-29, 23)

New centroid G' = (36 + 16-29)/3, (18+53+23)/3 = color(brown)((23/3, 94/3)

Distance moved by centroid is

$\vec{G G '} = \sqrt{{\left(\frac{7}{3} - \frac{23}{3}\right)}^{2} + {\left(3 - \frac{94}{3}\right)}^{2}} \approx \textcolor{g r e e n}{28.83}$