A line segment has endpoints at #(3 , 1)# and #(2 ,3)#. If the line segment is rotated about the origin by #pi #, translated horizontally by #4#, and reflected about the x-axis, what will the line segment's new endpoints be?

2 Answers
Aviv S.
Jan 28, 2018

The new endpoints are #(1,1)# and #(2,3)#.

Explanation:

Let's apply each of the transformations to both points, one at a time:

#pi# rotation around the origin (180º):
#(x,y)=>(-x,-y)#
#(3,1)=>(-3,-1)#
#(2,3)=>(-2,-3)#

Horizontal translation by #4#:
#(x,y)=>(x+4,y)#
#(-3,-1)=>(1,-1)#
#(-2,-3)=>(2,-3)#

Reflection over the #x#-axis:
#(x,y)=>(x,-y)#
#(1,-1)=>(1,1)#
#(2,-3)=>(2,3)#

The new endpoints are #(1,1)# and #(2,3)#.

Jim G.
Jan 28, 2018

#(1,1)" and "((2,3)#

Explanation:

#"since there are 3 transformations to be performed here"#

#"label the endpoints "A(3,1)" and "B(2,3)#

#color(blue)"First transformation"#

#"under a rotation about the origin by "pi#

#• " a point "(x,y)to(-x,-y)#

#rArrA(3,1)toA'(-3,-1)#

#rArrB(2,3)toB'(-2,-3)#

#color(blue)"Second transformation"#

#"under a translation "((4),(0))#

#• " a point "(x,y)to(x+4,y)#

#rArrA'(-3,-1)toA''(1,-1)#

#rArrB'(-2,-3)toB''(2,-3)#

#color(blue)"Third transformation"#

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

#• " a point "(x,y)to(x,-y)#

#rArrA''(1,-1)toA'''(1,1)#

#rArrB''(2,-3)toB'''(2,3)#

#"after all 3 transformations"#

#(3,1)to(1,1)" and "(2,3)to(2,3)#

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