Question

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?

Answer 1

(-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)
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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)"----------------------------------------------------"`