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

Apr 12, 2018

color(purple)("Distances of A & B after dilation and reflection "

color(green)(9.4868, 1.4142 " respy."

#### Explanation:

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

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

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

"Reflection Rule : reflect thru " x = 1, y = -3; h=1, k= -3; (2h-x, 2k-y)

$A ' ' \left(x , y\right) = A ' \left(\begin{matrix}2 h - x \\ 2 k - y\end{matrix}\right) = \left(2 - 5 , - 6 - 3\right) = \left(- 3 , - 9\right)$

$B ' ' \left(x , y\right) = B ' \left(\begin{matrix}2 h - x \\ 2 k - y\end{matrix}\right) = \left(2 - 1 , - 6 + 5\right) = \left(1 , - 1\right)$

$O A ' ' = \sqrt{- {3}^{2} + - {9}^{2}} = 9.4868$

$O B ' ' = \sqrt{{1}^{2} + - {1}^{2}} = 1.4142$