# 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?

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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

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

A translation of 4 units vertically maps:

A reflection in the x axis maps:

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

Putting these together in order:

Naming endpoints A and B:

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