# A line segment has endpoints at (5 ,9 ) and (2 ,2 ). The line segment is dilated by a factor of 2  around (5 ,3 ). What are the new endpoints and length of the line segment?

Jun 8, 2018

color(maroon)("New endpoints are" A' (5,21), B' (-4,0)

color(blue)("Length of the line segment "bar(A'B') = 15,23

#### Explanation:

$A \left(5 , 9\right) , B \left(2 , 2\right) , C \left(5 , 3\right) , \text{ dilation factor } 2$

$A ' C = 2 \left(a - c\right)$, $A ' - c = 2 \left(a - c\right)$ or $A ' = 2 a - c$

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 2 \left(\begin{matrix}5 \\ 9\end{matrix}\right) - \left(\begin{matrix}5 \\ 3\end{matrix}\right) = \left(\begin{matrix}5 \\ 15\end{matrix}\right)$

$B ' C = 2 \left(b - c\right)$, $B ' - c = 2 \left(b - c\right)$ or $A ' = 2 b - c$

$B ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 2 \left(\begin{matrix}2 \\ 2\end{matrix}\right) - \left(\begin{matrix}5 \\ 3\end{matrix}\right) = \left(\begin{matrix}- 1 \\ 1\end{matrix}\right)$

Length of the line segment

$\overline{A ' B '} = \sqrt{{\left(5 + 1\right)}^{2} + {14}^{2}} = 15.23$