A line segment has endpoints at #(3 ,4 )# and #(2 ,5 )#. If the line segment is rotated about the origin by # pi #, translated horizontally by # 2 #, and reflected about the x-axis, what will the line segment's new endpoints be?

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Answer 1 · 2016-10-20T16:10:38

#(-1, 4) and (0, 5)#

There is a special shortcut to rotate by #pi# but I will do that rotation the long way, because it shows how do it for any angle.

The first point rotated by #pi#:

#r = sqrt(3^2 + 4^2)#

#r = 5#

#theta = tan^-1(4/3) + pi#

#(5cos(tan^-1(4/3) + pi), 5sin(tan^-1(4/3) + pi)) =#

#(-3, -4)#

The second point rotated by #pi#:

#r = sqrt(2^2 + 5^2)#

#r = sqrt(29)#

#theta = tan^-1(5/2) + pi#

#(sqrt(29)cos(tan^-1(5/2) + pi), 5sin(tan^-1(5/2) + pi)) =#

#(-2, -5)#

Translated horizontally by 2 means add two to the x coordinates:

#(-3, -4) -> (-1, -4)#
#(-2, -5) -> (0, -5)#

Reflected about the x axis means multiply the y coordinates by -1:

#(-1, -4) ->(-1, 4)#
#(0, -5) -> (0,5)#