# A line segment has endpoints at (0 ,9 ) and (8 ,2 ). The line segment is dilated by a factor of 2  around (1 ,1 ). What are the new endpoints and length of the line segment?

Feb 12, 2018

New end points color(purple)(((-1),(17)),((15),(3))

Length of the line segment $\approx \textcolor{g r e e n}{21.26}$

#### Explanation:

$A \left(\begin{matrix}0 \\ 9\end{matrix}\right) , B \left(\begin{matrix}8 \\ 2\end{matrix}\right) , C \left(\begin{matrix}1 \\ 1\end{matrix}\right)$ Dilation factor around C is 2

$\vec{A ' C} = 2 \vec{A C} \mathmr{and} a ' = 2 a - c$

$a ' = 2 \left(\begin{matrix}0 \\ 9\end{matrix}\right) - \left(\begin{matrix}1 \\ 1\end{matrix}\right) = \left(\begin{matrix}- 1 \\ 17\end{matrix}\right)$

$\vec{B ' C} = 2 \left(B C\right) \mathmr{and} b ' = 2 b - c$

$b ' = 2 \left(\begin{matrix}8 \\ 2\end{matrix}\right) - \left(\begin{matrix}1 \\ 1\end{matrix}\right) = \left(\begin{matrix}15 \\ 3\end{matrix}\right)$

$\vec{A ' B '} = \sqrt{{\left(15 + 1\right)}^{2} + {\left(3 - 17\right)}^{2}} \approx \textcolor{g r e e n}{21.26}$