# A line segment has endpoints at (75 ,8 ) and (2 , 1). The line segment is dilated by a factor of 2  around (4 , 5). What are the new endpoints and length of the line segment?

Jul 14, 2018

color(crimson)("New coordinates are " (146,11), (0, -3)

color(crimson)("Length of the line segment '" = sqrt((146-0)^2 + (11--3)^2) ~~ 146.6697

#### Explanation:

$A \left(75 , 8\right) , B \left(2 , 1\right) , \text{ about point " D (4,5), " dilation factor } 2$

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = \left(2\right) a - \left(1\right) d = \left(2\right) \cdot \left(\begin{matrix}75 \\ 8\end{matrix}\right) - \left(1\right) \cdot \left(\begin{matrix}4 \\ 5\end{matrix}\right) = \left(\begin{matrix}146 \\ 11\end{matrix}\right)$

$B ' \left(\begin{matrix}x \\ y\end{matrix}\right) = \left(2\right) b - \left(1\right) d = \left(2\right) \cdot \left(\begin{matrix}2 \\ 1\end{matrix}\right) - \left(1\right) \cdot \left(\begin{matrix}4 \\ 5\end{matrix}\right) = \left(\begin{matrix}0 \\ - 3\end{matrix}\right)$

color(crimson)("New coordinates are " (146,11), (0, -3)

color(crimson)("Length of the line segment '" = sqrt((146-0)^2 + (11--3)^2) ~~ 146.6697