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

Apr 12, 2018

color(violet)("Distance of A & B from origin after reflection and dilation " 9, 12.04 " respy."

#### Explanation:

$A \left(1 , 2\right) , B \left(4 , 1\right) , \text{ reflected across } x = - 3 , y = 1$

$\text{Reflection Rule : reflect thru } x = - 3 , y = 1 , h = - 3 , k = 1$

$A ' \left(x , y\right) = A \left(2 h - x , 2 k - y\right) = \left(- 6 - 1 , 4 - 2\right) = \left(- 7 , 2\right)$

$B ' \left(x , y\right) = B \left(2 h - x , 2 k - y\right) = \left(- 6 - 4 , 4 - 1\right) = \left(- 10 , 3\right)$

Points A' & B' dilated about C(2,2) by a factor of 2.

$A ' \left(x , y\right) \to A ' ' \left(x , y\right) = A ' \left(x , y\right) - C \left(x , y\right) = \left(\left(- 7 , 2\right) - \left(2 , 2\right)\right) = \left(- 9 , 0\right)$

$B ' \left(x , y\right) \to B ' ' \left(x , y\right) = B ' \left(x , y\right) - C \left(x , y\right) = \left(\left(- 10 , 3\right) - \left(2 , 2\right)\right) = \left(- 12 , 1\right)$

OA' = 9, OB' = sqrt(-12^2 + 1^2 = 12.04