A line segment has endpoints at (3 ,4 ) and (2 ,5 ). If the line segment is rotated about the origin by  pi , translated horizontally by  2 , and reflected about the x-axis, what will the line segment's new endpoints be?

Oct 20, 2016

$\left(- 1 , 4\right) \mathmr{and} \left(0 , 5\right)$

Explanation:

There is a special shortcut to rotate by $\pi$ but I will do that rotation the long way, because it shows how do it for any angle.

The first point rotated by $\pi$:

$r = \sqrt{{3}^{2} + {4}^{2}}$

$r = 5$

$\theta = {\tan}^{-} 1 \left(\frac{4}{3}\right) + \pi$

$\left(5 \cos \left({\tan}^{-} 1 \left(\frac{4}{3}\right) + \pi\right) , 5 \sin \left({\tan}^{-} 1 \left(\frac{4}{3}\right) + \pi\right)\right) =$

$\left(- 3 , - 4\right)$

The second point rotated by $\pi$:

$r = \sqrt{{2}^{2} + {5}^{2}}$

$r = \sqrt{29}$

$\theta = {\tan}^{-} 1 \left(\frac{5}{2}\right) + \pi$

$\left(\sqrt{29} \cos \left({\tan}^{-} 1 \left(\frac{5}{2}\right) + \pi\right) , 5 \sin \left({\tan}^{-} 1 \left(\frac{5}{2}\right) + \pi\right)\right) =$

$\left(- 2 , - 5\right)$

Translated horizontally by 2 means add two to the x coordinates:

$\left(- 3 , - 4\right) \to \left(- 1 , - 4\right)$
$\left(- 2 , - 5\right) \to \left(0 , - 5\right)$

Reflected about the x axis means multiply the y coordinates by -1:

$\left(- 1 , - 4\right) \to \left(- 1 , 4\right)$
$\left(0 , - 5\right) \to \left(0 , 5\right)$