A line segment goes from (2 ,3 ) to (4 ,1 ). The line segment is dilated about (0 ,1 ) by a factor of 3. 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?

Aug 7, 2018

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

color(green)(sqrt 85, sqrt73 " respectivelyy."

Explanation:

$A \left(2 , 3\right) , B \left(4 , 1\right) , \text{ dilated by factor 3 about } C \left(0 , 1\right)$

$A \left(x , y\right) \to A ' \left(x , y\right) = 3 \cdot A \left(x , y\right) - 2 \cdot C \left(x , y\right) = \left(3 \cdot \left(2 , 3\right) - 2 \cdot \left(0 , 1\right)\right) = \left(6 , 7\right)$

$B \left(x , y\right) \to B ' \left(x , y\right) = 3 \cdot B \left(x , y\right) - 2 \cdot C \left(x , y\right) = \left(3 \cdot \left(4 , 1\right) - 2 \cdot \left(0 , 1\right)\right) = \left(12 , 1\right)$

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

$A ' ' \left(x , y\right) = A ' \left(\begin{matrix}2 h - x \\ 2 k - y\end{matrix}\right) = \left(4 - 6 , - 2 - 7\right) = \left(- 2 , - 9\right)$

$B ' ' \left(x , y\right) = B ' \left(\begin{matrix}2 h - x \\ 2 k - y\end{matrix}\right) = \left(4 - 12 , - 2 - 1\right) = \left(- 8 , - 3\right)$

$O A ' ' = \sqrt{- {2}^{2} + - {9}^{2}} = \sqrt{85}$

$O B ' ' = \sqrt{- {8}^{2} + - {3}^{2}} = \sqrt{73}$