# A line segment has endpoints at (8 ,4 ) and (3 ,7 ). The line segment is dilated by a factor of 1/2  around (3 ,2 ). What are the new endpoints and length of the line segment?

Jul 14, 2018

color(purple)("New end points are " (11/2, 3), (3,9/2)

color(purple)("Length of the line segment '" = sqrt((11/2- 3)^2 + (3-9/2)^2) ~~ 2.9155

#### Explanation:

$A \left(8 , 4\right) , B \left(3 , 7\right) , \text{ about point " D (3,2), " dilation factor } \frac{1}{2}$

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\frac{1}{2}\right) a - \left(- \frac{1}{2}\right) d = \left(\frac{1}{2}\right) \cdot \left(\begin{matrix}8 \\ 4\end{matrix}\right) - \left(- \frac{1}{2}\right) \cdot \left(\begin{matrix}3 \\ 2\end{matrix}\right) = \left(\begin{matrix}\frac{11}{2} \\ 3\end{matrix}\right)$

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

color(purple)("New end points are " (11/2, 3), (3,9/2)

color(purple)("Length of the line segment '" = sqrt((11/2- 3)^2 + (3-9/2)^2) ~~ 2.9155