A line segment goes from (1 ,1 ) to (4 ,2 ). The line segment is reflected across x=2, reflected across y=-1, and then dilated about (1 ,1 ) by a factor of 2. How far are the new endpoints from the origin?

Apr 12, 2018

color(purple)("Distance of A from origin after reflection and dilation " = 8.6

color(purple)("Distance of B from origin after reflection and dilation " = 9.06

Explanation:

$A \left(1 , 1\right) , B \left(4 , 2\right) \text{ reflected across " x = 2, y = -1 " in that order}$

$\text{Reflection rule : reflect thru } x = 2 , y = - 1 , h = 2 , k = - 1. \left(2 h - x , 2 k - y\right)$

$A ' \left(x , y\right) = \left(2 h - x , 2 k - y\right) = \left(4 - 1 , - 2 - 1\right) = \left(3 , - 3\right)$

$B ' \left(x , y\right) = \left(2 h - x , 2 k - y\right) = \left(4 - 4 , - 2 - 2\right) = \left(0 , - 4\right)$

$\text{A', B' dilated about C (1,1) by a factor of 2}$

$A ' \left(x , y\right) \to A ' ' \left(x , y\right) = 2 \cdot A ' \left(x ,\right) - C \left(x , y\right) = \left(\left(6 , - 6\right) - \left(1 , 1\right)\right) = \left(5 , - 7\right)$

$B ' \left(x , y\right) \to B ' ' \left(x , y\right) = 2 \cdot B ' \left(x , y\right) - C \left(x , y\right) = \left(\left(0 , - 8\right) - \left(1 , 1\right)\right) = \left(- 1 , - 9\right)$

$O A ' ' = \sqrt{{5}^{2} + {7}^{2}} = 8.6$

$O B ' ' = \sqrt{{15}^{2} + {9}^{2}} = 9.06$

color(purple)("Distance of A from origin after reflection and dilation " = 8.6

color(purple)("Distance of B from origin after reflection and dilation " = 9.06