A line segment has endpoints at $(2, 0)$ $(2 ,0 )$ and $(2, 1)$ $(2 ,1 )$ . If the line segment is rotated about the origin by $\frac{π}{2}$ $( pi)/2$ , translated vertically by $- 8$ $-8$ , and reflected about the x-axis, what will the line segment's new endpoints be?

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Answer 1 · 2018-06-11T12:26:42

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

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

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

Let $A=(2,0), B=(2,1)$

A rotation about the origin by $pi/2$ radians maps:

$(x,y)->(-y,x)$

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

$(x,y)->(x,y-8)$

A reflection in the x axis maps:

$(x,y)->(x,-y)$

We can put all these mappings together:

$(x,y)->(-y,x)->(-y,x-8)->(-y,-(x-8))$

$A->A'=(2,0)->(-(0),-(2-8))=(0,6)$

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

PLOT:

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A line segment has endpoints at (2 ,0 )(2,0)(2 ,0 ) and (2 ,1 )(2,1)(2 ,1 ). If the line segment is rotated about the origin by ( pi)/2 π2( pi)/2 , translated vertically by -8 −8-8 , and reflected about the x-axis, what will the line segment's new endpoints be?