# A line segment has endpoints at #(4 ,1 )# and #(5 ,3)#. If the line segment is rotated about the origin by #pi #, translated vertically by #-2 #, and reflected about the x-axis, what will the line segment's new endpoints be?

##### 1 Answer

(-4 , 3) , (-5 , 5)

#### Explanation:

under a rotation of

# pi# " about the origin " #a point (x , y) → (-x , -y)

Name the points A(4 , 1 ) and B(5 , 3)

#color(red)"----------------------------------------------"#

Step 1:A(4 , 1) → A'(-4 ,-1) and B(5 , 3) → B'(-5 , -3)

#color(red)"----------------------------------------------"# Under a translation of

#((0),(-2))# a point (x , y) → (x , y-2)

#color(red)"-----------------------------------------------"#

Step 2:A'(-4 , -1) →A'' (-4 , -3) and B'(-5 ,-3) → B''(-5 , -5)

#color(red)"--------------------------------------------------"# Under a reflection in the x-axis

a point (x , y) → (x , -y)

#color(red)"-------------------------------------------------"#

Step 3:A''(-4 , -3) → A'''(-4 , 3) and B''(-5 , -5) → (-5 , 5)

#color(red)"----------------------------------------------------"#