# A triangle has corners at (3 ,1 ), (7 ,-5 ), and (-4 ,-2 ). If the triangle is dilated by a factor of 5  about point (7 ,-6 ), how far will its centroid move?

Feb 5, 2018

Distance moved by centroid

$\vec{G G '} \approx \textcolor{g r e e n}{26.54}$ rounded to two decimals

#### Explanation:

Given : A (3,1), B (7,-5), C(-4,-2)

Dilated about D(7,-6) and dilation factor 5

To find distance the centroid has moved

$C e n t r o i d$G = (3+7+(-4))/3, (1-5-2)/3 = color(brown)((2,-2)

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

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

$\implies 5 \left(\begin{matrix}3 \\ 1\end{matrix}\right) - 4 \left(\begin{matrix}- 4 \\ - 2\end{matrix}\right) = \left(\begin{matrix}15 \\ 5\end{matrix}\right) - \left(\begin{matrix}- 16 \\ - 8\end{matrix}\right) = \left(\begin{matrix}31 \\ 13\end{matrix}\right)$

color(blue)(A' (31, 13)

Similarly,

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

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

$\implies 5 \left(\begin{matrix}7 \\ - 5\end{matrix}\right) - 4 \left(\begin{matrix}- 4 \\ - 2\end{matrix}\right) = \left(\begin{matrix}35 \\ - 25\end{matrix}\right) - \left(\begin{matrix}- 16 \\ - 8\end{matrix}\right) = \left(\begin{matrix}51 \\ 17\end{matrix}\right)$

color(blue)(B' (51, 17)

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

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

$\implies 5 \left(\begin{matrix}- 4 \\ - 2\end{matrix}\right) - 4 \left(\begin{matrix}- 4 \\ - 2\end{matrix}\right) = \left(\begin{matrix}- 20 \\ - 10\end{matrix}\right) - \left(\begin{matrix}- 16 \\ - 8\end{matrix}\right) = \left(\begin{matrix}- 4 \\ - 2\end{matrix}\right)$

color(blue)(C' (-4, -2)

New centroid G' = (31 + 51-4)/3, (13+17-2)/3 = color(brown)((26, 28/3)

Distance moved by centroid is

$\vec{G G '} = \sqrt{{\left(2 - 26\right)}^{2} + {\left(- 2 - \left(\frac{28}{3}\right)\right)}^{2}} \approx \textcolor{g r e e n}{26.54}$ rounded to two decimals