# A line segment has endpoints at (5 ,8 ) and (2 , 1). The line segment is dilated by a factor of 1/2  around (3 , 5). What are the new endpoints and length of the line segment?

Jul 14, 2018

color(brown)("New end points are " (4,13/2), (5/2, 3)

color(brown)("Length of the line segment '" = sqrt((4- 5/2)^2 + (13/2-3)^2) ~~ 3.8079

#### Explanation:

$A \left(5 , 8\right) , B \left(2 , 1\right) , \text{ about point " D (3,5), " dilation factor } \frac{1}{2}$

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\frac{1}{2}\right) a - \left(- \frac{1}{2}\right) d = \left(\frac{1}{2}\right) \cdot \left(\begin{matrix}5 \\ 8\end{matrix}\right) - \left(- \frac{1}{2}\right) \cdot \left(\begin{matrix}3 \\ 5\end{matrix}\right) = \left(\begin{matrix}4 \\ \frac{13}{2}\end{matrix}\right)$

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

color(brown)("New end points are " (4,13/2), (5/2, 3)

color(brown)("Length of the line segment '" = sqrt((4- 5/2)^2 + (13/2-3)^2) ~~ 3.8079