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?

1 Answer
Somebody N.
May 22, 2018

#(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)#

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