A line segment has endpoints at #(5 ,8 )# and #(7 ,4)#. If the line segment is rotated about the origin by #pi #, translated horizontally by #-1 #, and reflected about the y-axis, what will the line segment's new endpoints be?

1 Answer
mason m
Jan 21, 2016

#(6,-8),(8,-4)#

Explanation:

We can create a rule for this transformation.

Assume the original endpoints of the segment can be described as #color(red)((x,y)#.

The first way in which the segment is manipulated is with a rotation of #pi#, or #180^@#. This translates by taking the opposite of the #x# and #y# values of the point, or our new "rule" for this point in the transformation: #color(red)((-x,-y)#

The endpoints are then translated horizontally by #-1#. Horizontal movement corresponds to the #x# coordinate, whereas vertical corresponds to the #y# coordinate. Since this has been horizontally shifted, the new rule, working off the most previous rule, is: #color(red)((-x-1,-y)#

The final transformation is a reflection over the #y# axis, which means that the #x# coordinate's sign is flipped: #color(red)((x+1,-y)#

This is the rule for the entire transformation. Apply it to the points #(5,8)# and #(7,4)#.

#(5,8)rarr(6,-8)#

#(7,4)rarr(8,-4)#

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