A line segment has endpoints at #(4 ,0 )# and #(2 ,9 )#. If the line segment is rotated about the origin by #( pi)/2 #, translated vertically by #-8 #, and reflected about the x-axis, what will the line segment's new endpoints be?

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Answer 1 · 2016-10-11T04:40:07

The endpoints are #(0, 4)# and #(-9, 6)#

Rotate #(4, 0) by pi/2#:

#|4,0| = 4#

#theta = tan^-1(0/4) = 0#

The rotated angle

#theta_r = pi/2#

#(4cos(pi/2), 4sin(pi/2)) = (0, 4)#

Rotate #(2, 9) by pi/2#:

#|2,9| = sqrt(2^2 + 9^2)#

#|2,9| = sqrt(85)#

#theta = tan^-1(9/2)#

The rotated angle

#theta_r = tan^-1(9/2) + pi/2#

#(sqrt(85)cos(tan^-1(9/2) + pi/2), sqrt(85)sin(tan^-1(9/2) + pi/2)) = (-9, 2)#

Translate both points vertically by -8:
#(0, 4-8) = (0, -4)#
#(-9, 2-8) = (-9, -6)#

Reflect about the x axis:

#(0, -1*-4) = (0, 4)#
#(-9, -1*-6) = (-9, 6)#