Question

A line segment has endpoints at `(6 ,5 )` and `(8 ,7 )`. If the line segment is rotated about the origin by `pi`, translated horizontally by `2`, and reflected about the x-axis, what will the line segment's new endpoints be?

Answer 1

`color(violet)("After all 3 transformations " color(green)((7, 4)to(-4, -5)" and "(-7, -8)to(-7, 8)`

Explanation:

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

`A(6,5)" and "B(8,7)`

`color(maroon)"transformation of rotation about the origin of "pi`

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

`rArrA(6, 5)toA'(-6, -5)`

`rArrB(8, 7)toB'(-7, -8)`

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

`• " a point "(x,y)to(x + 2, y)`

`rArrA'(-6, -5) to A''(-4, -5)`

`rArrB'(-7, -8) to B''(-5, -8)`

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

`• " a point "(x,y)to(x,-y)`

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

`rArrB''(-7, -8)toB'''(-7, 8)`

`color(violet)("After all 3 transformations " color(green)((7, 4)to(-4, -5)" and "(-7, -8)to(-7, 8)`