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

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Feb 10, 2018

Distance moved by centroid $\vec{G G '} \approx 8.11$

#### Explanation:

Dilation factor 5, dilated about D (-1,-2)

Centroid $G = \frac{1 + 3 - 1}{3} , \frac{5 - 9 - 4}{3} \implies \left(\begin{matrix}1 \\ - \frac{8}{3}\end{matrix}\right)$

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

$G ' \left(\begin{matrix}x + 1 \\ y + 2\end{matrix}\right) = 5 \cdot \left(\begin{matrix}1 + 1 \\ - \frac{8}{3} + 2\end{matrix}\right) \implies \left(\begin{matrix}10 \\ - \frac{10}{3}\end{matrix}\right)$

$G ' \left(\begin{matrix}9 \\ - \frac{4}{3}\end{matrix}\right)$

$\vec{G ' G} = \sqrt{{\left(9 - 1\right)}^{2} + {\left(- \frac{4}{3} + \frac{8}{3}\right)}^{2}} \approx 8.11$

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