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

Apr 12, 2018

color(indigo)("Distances of A and B from origin after dilation and reflection "

color(indigo)(3.16, 12.65 " respectively"

#### Explanation:

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

$A ' \left(x , y\right) = 3 \cdot A \left(x , y\right) - C \left(x , y\right) = \left(\left(9 , 3\right) - \left(2 , 2\right)\right) = \left(7 , 1\right)$

$B ' \left(x , y\right) = {3}^{B} \left(x , y\right) - C \left(x , y\right) = \left(\left(6 , 12\right) - \left(2 , 2\right)\right) = \left(4 , 10\right)$

color(brown)("reflect thru " x = 4, y = -1, h=4, k= -1. (2h-x, 2k-y)"

$A ' ' \left(x , y\right) = \left(2 h - x , 2 k - y\right) = \left(\begin{matrix}8 - 7 \\ - 2 - 1\end{matrix}\right) = \left(1 , - 3\right)$

$\text{similarly } B ' ' \left(x , y\right) = \left(\begin{matrix}8 - 4 \\ - 2 - 10\end{matrix}\right) = \left(4 , - 12\right)$

$\overline{A ' ' O} = \sqrt{{1}^{2} + {3}^{2}} = \sqrt{10} = 3.16$

$\overline{B ' ' O} = \sqrt{{4}^{2} + {12}^{2}} = \sqrt{160} = 12.65$