# A line segment goes from (6 ,5 ) to (7 ,3 ). The line segment is dilated about (2 ,1 ) by a factor of 3. Then the line segment is reflected across the lines x = 3 and y=-4, in that order. How far are the new endpoints form the origin?

Apr 12, 2018

Distance of the points from the origin after dilation and reflection :

color(crimson)(bar(OA''') = 11.66, bar(OB''') = 20.62

#### Explanation:

Given : $A \left(6 , 5\right) , B \left(7 , 3\right)$

Now we have to find A', B' after rotation about point C (2,1) with a dilation factor of 3.

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

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 3 \cdot \left(\begin{matrix}6 \\ 5\end{matrix}\right) - \left(\begin{matrix}2 \\ 1\end{matrix}\right) = \left(\begin{matrix}16 \\ 14\end{matrix}\right)$

Similarly, $B ' \left(\begin{matrix}x \\ y\end{matrix}\right) = 3 \cdot \left(\begin{matrix}7 \\ 3\end{matrix}\right) - \left(\begin{matrix}2 \\ 1\end{matrix}\right) = \left(\begin{matrix}19 \\ 8\end{matrix}\right)$

Now to find

Reflection Rules :

color(crimson)("reflect over a line. ex: x=6, (2h-x, y)"

$\text{Reflected across } x = 3$

$A ' \left(x , y\right) \to A ' ' \left(x , y\right) = \left(\begin{matrix}6 - 16 \\ 14\end{matrix}\right) = \left(- 10 , 14\right)$

$B ' \left(x , y\right) \to B ' ' \left(x , y\right) = \left(\begin{matrix}2 \cdot 3 - 19 \\ 8\end{matrix}\right) = \left(- 13 , 8\right)$

$\text{Reflected across } y = - 4$

color(crimson)("reflect over a line. ex: y= -3, (x, 2k-y)"

A''(x,y) -> A'''(x,y) = ((-10, 8-14) = (-10,6)

$B ' ' \left(x , y\right) \to B ' ' ' \left(x , y\right) = \left(- 13 , - 8 - 8\right) = \left(- 13 , - 16\right)$

$\overline{O A ' ' '} = \sqrt{- {10}^{2} + {6}^{2}} = 11.66$

$\overline{O B ' ' '} = \sqrt{- {13}^{2} + - {16}^{2}} = 20.62$