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

Feb 12, 2018

Centroid has moved by a distance of $\approx \textcolor{g r e e n}{26.36}$

#### Explanation:

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

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

To find the distance, centroid has moved

$C e n t r o i d$G = (1+3+(-1))/3, (3-2+7)/3 = color(brown)((1,8/3)

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

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

$\implies 5 \left(\begin{matrix}1 \\ 3\end{matrix}\right) - 4 \left(\begin{matrix}- 2 \\ - 1\end{matrix}\right) = \left(\begin{matrix}5 \\ 15\end{matrix}\right) - \left(\begin{matrix}- 8 \\ - 4\end{matrix}\right) = \left(\begin{matrix}- 3 \\ 19\end{matrix}\right)$

color(blue)(A' (-3, 19)

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

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

$\implies 5 \left(\begin{matrix}3 \\ - 2\end{matrix}\right) - 4 \left(\begin{matrix}- 2 \\ - 1\end{matrix}\right) = \left(\begin{matrix}15 \\ - 10\end{matrix}\right) - \left(\begin{matrix}- 8 \\ - 4\end{matrix}\right) = \left(\begin{matrix}51 \\ 17\end{matrix}\right)$

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

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

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

$\implies 5 \left(\begin{matrix}- 1 \\ 7\end{matrix}\right) - 4 \left(\begin{matrix}- 2 \\ - 1\end{matrix}\right) = \left(\begin{matrix}- 5 \\ 35\end{matrix}\right) - \left(\begin{matrix}- 8 \\ - 4\end{matrix}\right) = \left(\begin{matrix}- 3 \\ 39\end{matrix}\right)$

color(blue)(C' (-3, 39)

New centroid G' = (-3 + 51-3)/3, (19+17+39)/3 = color(brown)((15, 25)#

Distance moved by centroid is

$\vec{G G '} = \sqrt{{\left(1 - 15\right)}^{2} + {\left(\frac{8}{3} - 25\right)}^{2}} \approx \textcolor{g r e e n}{26.36}$