Question

A line segment has endpoints at `(1 ,2 )` and `(9 ,4 )`. 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

`color(chocolate)("After all 3 transformations " color(maroon)((1, 2)to(-2, -2)" and "(9, 4)to(4, 6)`

Explanation:

`"since there are 3 transformations to be performed"`
`"label the endpoints"`

`A(1, 2)" and "B(9, 4)`

`color(purple)"transformation of rotation about the origin of "(3pi) /2`

`" a point "(x,y) to (y, -x)`

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

`rArrB(9, 4)toB'(4, -9)`

`color(purple)"next transformation under a vertical translation " ((0),(3))`

`• " a point "(x,y)to(x, y + 3)`

`rArrA'(2, -1) to A''(2, 2)`

`rArrB'(4, -9) to B''(4, -6)`

`color(purple)"last transformation under a reflection in the x -axis"`

`• " a point "(x,y)to(x, -y)`

`rArrA''(2, 2)toA'''(-2, -2)`

`rArrB''(4, -6)toB'''(4, 6)`

`color(chocolate)("After all 3 transformations " color(maroon)((1, 2)to(-2, -2)" and "(9, 4)to(4, 6)`

Answer 2

`(2,-2)" and "(4,6)`

Explanation:

`"Since there are 3 transformations to be performed label"`
`"the endpoints"`

`A=(1,2)" and "B=(9,4)`

`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(9,4)toB'(4,-9)`

`color(blue)"second transformation"`

`"under a vertical translation "((0),(3))`

`"a point "(x,y)to(x,y+3)`

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

`B'(4,-9)toB''(4,-6)`

`color(blue)"third transformation"`

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

`"a point "(x,y)to(x,-y)`

`A(2,2)toA'''(2,-2)`

`B''(4,-6)toB'''(4,6)`

`"After all 3 transformations"`

`(1,2)to(2,-2)" and "(9,4)to(4,6)`