Question

A line segment has endpoints at `(0 ,2 )` and `(3 ,5 )`. If the line segment is rotated about the origin by `( 3 pi)/2`, translated vertically by `3`, and reflected about the x-axis, what will the line segment's new endpoints be?

Answer 1

`(2,-3)" and " (5,0)`

Explanation:

Since there are 3 transformations to be performed here, name the endpoints A (0 ,2) and B (3 ,5) so that they may be 'tracked' after each transformation.

First transformation Under a rotation about the origin of `(3pi)/2`

a point (x ,y) → (y ,-x)

hence A(0 ,2) → A'(2 ,0) and B(3 ,5) → B'(5 ,-3)

Second transformation Under a translation `((0),(3))`

a point (x ,y) → (x ,y+3)

hence A'(2 ,0) → A''(2 ,3) and B'(5 ,-3) → B''(5 ,0)

Third transformation Under a reflection in the x-axis

a point (x ,y) → (x ,-y)

hence A''(2 ,3) → A'''(2 ,-3) and B''(5 ,0) → B'''(5 ,0)

Thus after all 3 transformations:

`(0,2)to(2,-3)" and " (3,5)to(5,0)`