# A line segment has endpoints at (4 ,0 ) and (2 ,9 ). If the line segment is rotated about the origin by ( pi)/2 , translated vertically by -8 , and reflected about the x-axis, what will the line segment's new endpoints be?

Oct 11, 2016

The endpoints are $\left(0 , 4\right)$ and $\left(- 9 , 6\right)$

#### Explanation:

Rotate $\left(4 , 0\right) b y \frac{\pi}{2}$:

$| 4 , 0 | = 4$

$\theta = {\tan}^{-} 1 \left(\frac{0}{4}\right) = 0$

The rotated angle

${\theta}_{r} = \frac{\pi}{2}$

$\left(4 \cos \left(\frac{\pi}{2}\right) , 4 \sin \left(\frac{\pi}{2}\right)\right) = \left(0 , 4\right)$

Rotate $\left(2 , 9\right) b y \frac{\pi}{2}$:

$| 2 , 9 | = \sqrt{{2}^{2} + {9}^{2}}$

$| 2 , 9 | = \sqrt{85}$

$\theta = {\tan}^{-} 1 \left(\frac{9}{2}\right)$

The rotated angle

${\theta}_{r} = {\tan}^{-} 1 \left(\frac{9}{2}\right) + \frac{\pi}{2}$

$\left(\sqrt{85} \cos \left({\tan}^{-} 1 \left(\frac{9}{2}\right) + \frac{\pi}{2}\right) , \sqrt{85} \sin \left({\tan}^{-} 1 \left(\frac{9}{2}\right) + \frac{\pi}{2}\right)\right) = \left(- 9 , 2\right)$

Translate both points vertically by -8:
$\left(0 , 4 - 8\right) = \left(0 , - 4\right)$
$\left(- 9 , 2 - 8\right) = \left(- 9 , - 6\right)$

Reflect about the x axis:

$\left(0 , - 1 \cdot - 4\right) = \left(0 , 4\right)$
$\left(- 9 , - 1 \cdot - 6\right) = \left(- 9 , 6\right)$