A line segment has endpoints at #(5 , 8)# and #(6 , 1)#. If the line segment is rotated about the origin by #pi/2 #, translated vertically by #-2#, and reflected about the y-axis, what will the line segment's new endpoints be?

2 Answers
Mr. Hsia
May 21, 2018

Rule for Rotation by #pi/2# or 90 degrees Counter-Clockwise:
#(x, y) -> (-y, x)#

Rule for Translated Vertically by #-2#
#(x, y) -> (x, y-2)#

Rule for Reflection across y-axis
#(x, y) -> (-x, y)#

In that same order:
#(5, 8) -> (-8, 5) -> (-8, 3) -> (8, 3)#
#(6, 1) -> (-1, 6) -> (-1, 4) -> (1, 4)#

To Check, each of these transformations are isometric which means the distance between the points will not change.

Original Points
#sqrt((5-6)^2 + (8-1)^2) = sqrt((-1)^2 + (7)^2) = sqrt(1 + 49) = sqrt (50)#

New Points
#sqrt((8-1)^2 + (3-4)^2) = sqrt((7)^2 + (-1)^2) = sqrt(49 + 1) = sqrt(50)#

They have the same distance so the segment endpoints are correct.

Jim G.
May 21, 2018

#(8,3)" and "(1,4)#

Explanation:

#"since there are 3 transformations to be performed"#
#"label the endpoints"#

#A=(5,8)" and "B=(6,1)#

#color(blue)"first transformation"#

#"under a rotation about the origin of "pi/2#

#• " a point "(x,y)to(-y,x)#

#rArrA(5,8)toA'(-8,5)#

#rArrB(6,1)toB'(-1,6)#

#color(blue)"second transformation"#

#"under a vertical translation "((0),(-2))#

#• " a point "(x,y)to(x,y-2)#

#rArrA'(-8,5)toA''(-8,3)#

#rArrB'(-1,6)toB''(-1,4)#

#color(blue)"third transformation"#

#"under a reflection in the y-axis"#

#• " a point "(x,y)to(-x,y)#

#rArrA''(-8,3)toA'''(8,3)#

#rArrB''(-1,4)toB'''(1,4)#

#"After all 3 transformations"#

#(5,8)to(8,3)" and "(6,1)to(1,4)#

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