# A triangle has corners at (-1 ,3 ), (3 ,-2 ), and (8 ,4 ). If the triangle is dilated by a factor of 5  about point (-2 ,6 ), how far will its centroid move?

Feb 5, 2018

Centroid moves by a distance of $\approx \textcolor{g r e e n}{24.5}$ units

#### Explanation:

Given A (-1,3), B(3,-2), C (8,4),

Dilation point D (-2,6), Dilation factor 5

To find
1. Present centroid,
2. centroid after dilation and
3. distance between the two centroids.

Present centroid G = (-1 + 3 + 8)/3, (3 - 2 + 4) /3 = color(brown)((10/3, 5/3)

$\overline{D A '} = 5 \cdot \left(\overline{D A}\right)$

$a ' - d = 5 a - 5 d$

$a ' = 5 a - 4 d = 5 \left(\begin{matrix}- 1 \\ 3\end{matrix}\right) - 4 \left(\begin{matrix}- 2 \\ 6\end{matrix}\right)$

$\implies \left(\begin{matrix}- 5 \\ 15\end{matrix}\right) - \left(\begin{matrix}- 8 \\ 24\end{matrix}\right) = \left(\begin{matrix}3 \\ - 9\end{matrix}\right)$

color(green)(A' (3, -9)

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

$b ' = 5 b - 4 d = 5 \left(\begin{matrix}3 \\ - 2\end{matrix}\right) - 4 \left(\begin{matrix}- 2 \\ 6\end{matrix}\right)$

$\implies \left(\begin{matrix}15 \\ - 10\end{matrix}\right) - \left(\begin{matrix}- 8 \\ 24\end{matrix}\right) = \left(\begin{matrix}23 \\ - 34\end{matrix}\right)$

color(green)(B'(23, -34)

Similarly, $\overline{D C '} = 5 \cdot \overline{D C}$

$c ' = 5 c - 4 d = 5 \left(\begin{matrix}8 \\ 4\end{matrix}\right) - 4 \left(\begin{matrix}- 2 \\ 6\end{matrix}\right)$

$\implies \left(\begin{matrix}40 \\ 20\end{matrix}\right) - \left(\begin{matrix}- 8 \\ 24\end{matrix}\right) = \left(\begin{matrix}48 \\ - 4\end{matrix}\right)$

color(green)(C' (48, -4)

Centroid of dilated triangle A'B'C' is

G' = (3 + 23 + 48) / 3, (-9-34-4)/3 = color(brown)((74/3, -47/3)#

Distance between centroids can be found using distance formula,

$\overline{G G '} = \sqrt{{\left(\left(\frac{74}{3}\right) - \left(\frac{10}{3}\right)\right)}^{2} + {\left(\left(- \frac{47}{3}\right) - \left(\frac{5}{3}\right)\right)}^{2}}$

$\overline{G G '} = \sqrt{{\left(\frac{64}{3}\right)}^{2} + {\left(\frac{52}{3}\right)}^{2}} \approx \textcolor{g r e e n}{24.5}$
(corrected to one decimal)