A line segment goes from `(1 ,2 )` to `(4 ,1 )`. The line segment is reflected across `x=-1`, reflected across `y=3`, and then dilated about `(2 ,2 )` by a factor of `3`. How far are the new endpoints from the origin?

Given points `A(1,2) , B (4,1)`

Reflected across `x = -1`, `y = 3`

`color(blue)("Reflection Rules :"`
`color(blue)("reflect over x-axis. (x,-y)"`
`color(blue)("reflect over y-axis. (-x,y)"`
`color(blue)("reflect over line y=x. (y,x)"`
`color(blue)("reflect over line y= -x. (-y,-x)"`
`color(blue)("reflect thru origin. (-x,-y)"`

`color(brown)("reflect thru a different point. ex: (5,-1) h=5 k= -1. (2h-x, 2k-y)"`

`color(blue)("reflect over a line. ex: x=6. (2h-x, y)"`
`color(blue)("reflect over a line. ex: y= -3. (x, 2k-y)"`


`A (x,y) -> A’(x, y) = (1,2) -> ((2x’- 1), (2y’ - 2))`

`A’(x,y) => ((-2 - 1), (6 - 2)) => (-3, 4)`

`B (x,y) - > B’ (x,y) = (4 , 1) -> ((2x’ - 4), (2y’ - 1)`

`B’(x,y) => ((-2 - 4), (6 - 1) => (-6, 5)`

New coordinates after reflection are `A’(-3, 4), B’(-6, 5)`

Now we we have to find A”, B” after rotation about point C (2,2) with a dilation factor of 3.

`A”(x, y) -> 3 * A’(x,y) - C (x,y)`

`A”((x),(y)) = 3 * ((-3), (4)) - ((2), (2)) = ((-11),(10))`

Similarly, `B”((x),(y)) = 2 * ((-6), (5)) - ((2), (2)) = ((-14),(8))`

New endpoints are `A”(-11, 10), B”(-14, 8)`

Distance of new points from origin

`vec(OA”) = sqrt(-11^2 + 10^2) ~~ 14.87`

`vec(OB”) = sqrt(-14^2 + 8^2) ~~ 16.12`