Question

A line segment has endpoints at `(4 ,7 )` and `(5 ,3 )`. If the line segment is rotated about the origin by `( 3 pi)/2`, translated horizontally by `- 2`, and reflected about the y-axis, what will the line segment's new endpoints be?

Answer 1

`color(brown)("After all 3 transformations " color(green)((4, 7)to(-5, -4)" and "(5, 3)to(-1, -5)`

Explanation:

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

`A(4, 7)" and "B(5, 3)`

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

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

`rArrA(4, 7)toA'(7, -4)`

`rArrB(5, 3)toB'(3, -5)`

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

`• " a point "(x,y)to(x - 2, y)`

`rArrA'(7, -4) to A''(5, -4)`

`rArrB'(3, -5) to B''(1, -5)`

`color(indigo)"last transformation under a reflection in the y -axis"`

`• " a point "(x,y)to(-x, y)`

`rArrA''(5, -4)toA'''(-5, -4)`

`rArrB''(1, -5)toB'''(-1, -5)`

`color(brown)("After all 3 transformations " color(green)((4, 7)to(-5, -4)" and "(5, 3)to(-1, -5)`

Answer 2

`(-5,-4)" and "(-1,-5)`

Explanation:

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

`A=(4,7)" and "B=(5,3)`

`color(blue)"first transformation"`

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

`• " a point "(x,y)to(y,-x)`

`A(4,7)toA'(7,-4)`

`B(5,3)toB'(3,-5)`

`color(blue)"second transformation"`

`"under a horizontal translation "((-2),(0))`

`• " a point "(x,y)to(x-2,y)`

`A'(7,-4)to(A''(5,-4)`

`B'(3,-5)toB''(1,-5)`

`color(blue)"third transformation"`

`"under a reflection about the y-axis"`

`• " a point "(x,y)to(-x,y)`

`A''(5,-4)toA'''(-5,-4)`

`B''(1,-5)toB'''(-1,-5)`

`"After all 3 transformations"`

`(4,7)to(-5,-4)" and "(5,3)to(-1,-5)`