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

Jul 14, 2018

color(violet)("New end points are " (3/2, 7/2), (5/2,3)

color(violet)("Length of the line segment '" = sqrt((3/2- 5/2)^2 + (7/2-3)^2) ~~ 1.118

#### Explanation:

$A \left(1 , 2\right) , B \left(3 , 1\right) , \text{ about point " D (2,5), " dilation factor } \frac{1}{2}$

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\frac{1}{2}\right) a - \left(- \frac{1}{2}\right) d = \left(\frac{1}{2}\right) \cdot \left(\begin{matrix}1 \\ 2\end{matrix}\right) - \left(- \frac{1}{2}\right) \cdot \left(\begin{matrix}2 \\ 5\end{matrix}\right) = \left(\begin{matrix}\frac{3}{2} \\ \frac{7}{2}\end{matrix}\right)$

B'((x),(y)) = (1/2)b - (-1/2)d = (1/2)* ((3),(1)) - (-1/2)*((2),(5)) = ((5/2),(3)

color(violet)("New end points are " (3/2, 7/2), (5/2,3)

color(violet)("Length of the line segment '" = sqrt((3/2- 5/2)^2 + (7/2-3)^2) ~~ 1.118