# A line segment has endpoints at #(1 ,2 )# and #(9 ,3 )#. If the line segment is rotated about the origin by #( 3 pi)/2 #, translated horizontally by # 2 #, and reflected about the x-axis, what will the line segment's new endpoints be?

##### 2 Answers

#color(brown)("After all 3 transformations " color(orange)((1, 2)to(4, 1)" and "(9, 3)to(5, 9)#

#### Explanation:

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

#"label the endpoints"#

#A(1, 2)" and "B(9, 3)#

#color(indigo)"transformation of rotation about the origin of "(3pi) / 2#

#rArrA(1, 2)toA'(2, -1)#

#rArrB(9, 3)toB'(3, -9)#

#color(indigo)"next transformation under a horizontal translation " ((2),(0))#

#• " a point "(x,y)to(x + 2, y)#

#rArrA'(2, -1) to A''(4, -1)#

#rArrB'(3, -9) to B''(5, -9)#

#color(indigo)"last transformation under a reflection in the x -axis"#

#• " a point "(x,y)to(x, -y)#

#rArrA''(4, -1)toA'''(4, 1)#

#rArrB''(5, -9)toB'''(5, 9)#

#color(brown)("After all 3 transformations " color(orange)((1, 2)to(4, 1)" and "(9, 3)to(5, 9)#

#### Explanation:

#"Since there are 3 transformations to be performed label"#

#"the endpoints"#

#A=(1,2)" and "B=(9,3)#

#color(blue)"first transformation"#

#"under a rotation about the origin of "(3pi)/2#

#• " a point "(x,y)to(y,-x)#

#A(1,2)toA'(2,-1)#

#B(9,3)toB'(3,-9)#

#color(blue)"second transformation"#

#"under a horizontal translation "((2),(0))#

#• " a point "(x,y)to(x+2,y)#

#A'(2,-1)toA''(4,-1)#

#B'(3,-9)toB''(5,-9)#

#color(blue)"third transformation"#

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

#• " a point "(x,y)to(x,-y)#

#A''(4,-1)toA'''(4,1)#

#B''(5,-9)toB'''(5,9)#

#"After all 3 transformations"#

#(1,2)to(4,1)" and "(9,3)to(5,9)#