Question

A line segment has endpoints at `(8 , 4)` and `(1 , 2)`. If the line segment is rotated about the origin by `(pi)/2`, translated vertically by `4`, and reflected about the x-axis, what will the line segment's new endpoints be?

Answer 1

`(4,-12)` and `(2,-5)`

Explanation:

You have not specified the direction of the rotation, so I will take this as being counter clockwise. If we look at each transformation and translation in order we get the following:

A rotation of `pi/2` anticlockwise will map:

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

This result takes a little thought. The easiest way to see this is, to think not of rotating a point, but rotating the axes themselves. If we rotate the axes `pi/2` counterclockwise, the positive x axis becomes the positive y axis and the positive y axis becomes the negative x axis.

A translation of 4 units vertically maps:

`(x,y)->(x,y+4)`

A reflection in the x axis maps:

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

This is the same as reflecting the axes, so positive y becomes negative y and x remains unchanged.

Putting these together in order:

`(x,y)->(-y,x)->(-y,x+4)->(y,-(x+4))`

Naming endpoints A and B:

`A=(8,4)`

`B=(1,2)`

`A=(8,4)->(y,-(x+4))=(4,-12)`

`B=(1,2)->(y,-(x+4))=(2,-5)`