# A line segment has endpoints at (1 ,2 ) and (3 , 9). The line segment is dilated by a factor of 1/2  around (4 , 2). What are the new endpoints and length of the line segment?

Jul 14, 2018

color(red)("New end points are " (5/2, 2), (7/2,11/2)

color(red)("Length of the line segment '" = sqrt((5/2- 7/2)^2 + (11/2-2)^2) ~~ 3.64

#### Explanation:

$A \left(1 , 2\right) , B \left(3 , 9\right) , \text{ about point " D (4,2), " dilation factor } \frac{1}{2}$

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

B'((x),(y)) = (1/2)b - (-1/2)d = (1/2)* ((3),(9)) - (-1/2)*((4),(2)) = ((7/2),(11/2)

color(red)("New end points are " (5/2, 2), (7/2,11/2)

color(red)("Length of the line segment '" = sqrt((5/2- 7/2)^2 + (11/2-2)^2) ~~ 3.64