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

Feb 12, 2018

Centroid will move by $\approx \textcolor{g r e e n}{10.75}$

#### Explanation:

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

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

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

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

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

$\implies 5 \left(\begin{matrix}- 7 \\ 3\end{matrix}\right) - 4 \left(\begin{matrix}1 \\ - 5\end{matrix}\right) = \left(\begin{matrix}- 35 \\ 15\end{matrix}\right) - \left(\begin{matrix}4 \\ - 20\end{matrix}\right) = \left(\begin{matrix}- 31 \\ - 5\end{matrix}\right)$

color(blue)(A' (-31, -5)

Similarly,

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

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

$\implies 5 \left(\begin{matrix}4 \\ - 1\end{matrix}\right) - 4 \left(\begin{matrix}1 \\ - 5\end{matrix}\right) = \left(\begin{matrix}20 \\ - 5\end{matrix}\right) - \left(\begin{matrix}4 \\ - 20\end{matrix}\right) = \left(\begin{matrix}16 \\ 15\end{matrix}\right)$

color(blue)(B' (16, 15)

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

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

=> 5((2,(1)) - 4((1),(-5)) = ((10),(5)) - ((4),(-20)) = ((6),(25))

color(blue)(C' (6, 25)

New centroid G' = (-31 + 16+6)/3, (-5+15+25)/3 = color(brown)((-3, 35/3)

Distance moved by centroid is

$\vec{G G '} = \sqrt{{\left(- \frac{1}{3} - 3\right)}^{2} + {\left(1 - \left(\frac{35}{3}\right)\right)}^{2}} \approx \textcolor{g r e e n}{10.75}$ rounded to two decimals