# A line segment has endpoints at (2 ,0 ) and (2 ,1 ). 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?

Jun 11, 2018

color(blue)(A'=(0,6)

color(blue)(B'=(-1,6)

#### Explanation:

No direction of rotation has been given, so I will take this as anti-clockwise.

Let $A = \left(2 , 0\right) , B = \left(2 , 1\right)$

A rotation about the origin by $\frac{\pi}{2}$ radians maps:

$\left(x , y\right) \to \left(- y , x\right)$

A translation by $- 8$ units in the vertical directions maps:

$\left(x , y\right) \to \left(x , y - 8\right)$

A reflection in the x axis maps:

$\left(x , y\right) \to \left(x , - y\right)$

We can put all these mappings together:

$\left(x , y\right) \to \left(- y , x\right) \to \left(- y , x - 8\right) \to \left(- y , - \left(x - 8\right)\right)$

$A \to A ' = \left(2 , 0\right) \to \left(- \left(0\right) , - \left(2 - 8\right)\right) = \left(0 , 6\right)$

B->B'=(2,1)->(-(1),-((2-8))=(-1,6)

PLOT: