Question

A line segment has endpoints at `(5 , 2)` and `(3 , 1)`. If the line segment is rotated about the origin by `pi/2`, translated vertically by `-2`, and reflected about the y-axis, what will the line segment's new endpoints be?

Answer 1

`color(blue)((-2,-7)`

`color(blue)((-1,-5)`

Explanation:

No direction of rotation is given, so I will take this as anti-clockwise:

A rotation of `pi/2` anti-clockwise maps:

`(x,y)->(y,-x)`

A translation of -2 units map:

`(x,y)->(x,y-2)`

A reflection in the y axis map:

`(x,y)->(-x,y)`

If we combine these, we get:

`(x,y)->(y,-x)->(y,-x-2)->(-y,-x-2)`

Therefore:

`(5,2)->(-y,-x-2)=(-2,-7)`

`(3,1)->(-y,-x-2)=(-1,-5)`