# A line segment goes from (2 ,3 ) to (4 ,1 ). The line segment is dilated about (1 ,1 ) by a factor of 2. Then the line segment is reflected across the lines x=2 and y=-1, in that order. How far are the new endpoints from the origin?

Apr 12, 2018

$\text{Distance of the two points from origin after dilation and reflection }$
$\textcolor{b l u e}{3.16 , 4.24}$

#### Explanation:

$A \left(2 , 3\right) , B \left(4 , 1\right) \text{ dilated about " C (1,1) " by a factor of 2}$

$A ' \left(x ,\right) = 2 \cdot A \left(x , y\right) - C \left(x , y\right) = \left(\left(4 , 6\right) - \left(1 , 1\right)\right) = \left(3 , 5\right)$

$B ' \left(x ,\right) = 2 \cdot B \left(x , y\right) - C \left(x , y\right) = \left(\left(8 , 2\right) - \left(1 , 1\right)\right) = \left(7 , 1\right)$

color(crimson)("reflect thru across " x=2, y = -1, h=2 k= -1. (2h-x, 2k-y)"

$A ' ' \left(x , y\right) = \left(2 \cdot 2 - 3\right) , \left(2 \cdot - 1 - 5\right) = \left(1 , - 3\right)$

$B ' ' \left(x , y\right) = \left(2 \cdot 2 - 7\right) , \left(2 \cdot - 1 - 1\right) = \left(- 3 , - 3\right)$

$A ' ' O = \sqrt{{1}^{2} + {3}^{3}} = 3.16$

$B ' ' O = \sqrt{{3}^{2} + {3}^{2}} = 4.24$