# A line segment has endpoints at (3 ,2 ) and (5 ,4 ). The line segment is dilated by a factor of 4  around (1 ,3 ). What are the new endpoints and length of the line segment?

Jul 14, 2018

color(brown)("New coordinates are " (9, -1), (17, 7)

color(brown)("Length of the line segment '" = sqrt((9-17)^2 + (-1-7)^2) ~~ 11.3137

#### Explanation:

$A \left(8 , 4\right) , B \left(3 , 2\right) , \text{ about point " D (1,3), " dilation factor } 4$

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = \left(4\right) a - \left(3\right) d = \left(4\right) \cdot \left(\begin{matrix}3 \\ 2\end{matrix}\right) - \left(3\right) \cdot \left(\begin{matrix}1 \\ 3\end{matrix}\right) = \left(\begin{matrix}9 \\ - 1\end{matrix}\right)$

$B ' \left(\begin{matrix}x \\ y\end{matrix}\right) = \left(4\right) b - \left(3\right) d = \left(4\right) \cdot \left(\begin{matrix}5 \\ 4\end{matrix}\right) - \left(3\right) \cdot \left(\begin{matrix}1 \\ 3\end{matrix}\right) = \left(\begin{matrix}17 \\ 7\end{matrix}\right)$

color(brown)("New coordinates are " (9, -1), (17, 7)

color(brown)("Length of the line segment '" = sqrt((9-17)^2 + (-1-7)^2) ~~ 11.3137