# A line segment has endpoints at (2 ,1 ) and (3 ,5 ). 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?

Jan 24, 2016

$P \left(1 , - 6\right) \mathmr{and} Q \left(5 - 5\right)$

#### Explanation:

Let $P \left(x , y\right)$ then $P ' \left(x ' , y '\right) = R \left(\theta\right) P \left(x , y\right)$
for $\theta = \frac{\pi}{2}$
$P ' \left(x ' , y '\right) = R \left(\frac{\pi}{2}\right) P \left(x , y\right) = P ' \left(- y , x\right)$ this a special case of the generalized rotation operation.

Now $P \left(2 , 1\right) \to P ' \left(- 1 , 2\right)$
and $Q \left(3 , 5\right) \to Q ' \left(- 5 , 3\right)$
To translate by -8 simply subtract 8 to y part
$P ' \left(- 1 , 2\right) \to P ' ' \left(- 1 , - 6\right)$
$Q ' \left(- 5 , 3\right) \to Q ' ' \left(- 5 , - 5\right)$
reflection about x axis will change the sign of point x
$P ' ' \left(- 1 , - 6\right) \to P ' ' ' \left(1 , - 6\right)$
$Q ' ' \left(- 5 , - 5\right) \to Q ' ' ' \left(5 - 5\right)$