# A line segment goes from (2 ,6 ) to (1 ,3 ). The line segment is dilated about (2 ,0 ) by a factor of 2. Then the line segment is reflected across the lines x = 2 and y=5, in that order. How far are the new endpoints from the origin?

Apr 12, 2018

color(green)("Dist of A from origin after dilation and reflections " = 2.83

color(green)("Dist of B from origin after dilation and reflections " = 5.66

#### Explanation:

$A \left(2 , 6\right) , B \left(1 , 3\right) , \text{ Dilated about " C(2,0) " by factor } 3$

$A ' \left(x , y\right) \to 2 \cdot A \left(x , y\right) - C \left(x , y\right) = \left(\left(4 , 12\right) - \left(2 , 0\right)\right) = \left(2 , 12\right)$

$B ' \left(x , y\right) \to 2 \cdot B \left(x , y\right) - C \left(x , y\right) = \left(\left(2 , 6\right) - \left(2 , 0\right)\right) = \left(0 , 6\right)$

Line segment A'B' reflected across x = 2, y = 5 in that order.

Reflection Rule :

color(crimson)("reflect over a line. ex: x=h. (2h-x, y)"

A'(x,y) -> A''(x,y) = (4 - 2, 12) -> (2,12)

$B ' \left(x , y\right) - > B ' ' \left(x , y\right) = \left(4 - 0 , 6\right) \to \left(4 , 6\right)$

color(crimson)("reflect over a line. ex: y= k. (x, 2k-y)"

$A ' ' \left(x , y\right) \to A ' ' ' \left(x , y\right) = \left(2 , 10 - 12\right) \to \left(2 , - 2\right)$

$B ' ' \left(x , y\right) \to B ' ' ' \left(x , y\right) = \left(4 , 10 - 6\right) \to \left(4 , 4\right)$

$\overline{A ' ' ' O} = \sqrt{{2}^{2} + {2}^{2}} = 2.83$

$\overline{B ' ' ' O} = \sqrt{{4}^{2} + {4}^{2}} = 5.66$

color(green)("Dist of A from origin after dilation and [reflections](https://socratic.org/geometry/transformations/reflections) " = 2.83

color(green)("Dist of B from origin after dilation and reflections " = 5.66#