Question

A line segment goes from `(1 ,2 )` to `(4 ,7 )`. The line segment is reflected across `x=6`, reflected across `y=-1`, and then dilated about `(1 ,1 )` by a factor of `2`. How far are the new endpoints from the origin?

Answer 1

Original segment `A_0B_0`, where `A_0=(1,2), B_0=(4,7)`,
is transformed into `AB`, where `A=(21,-9), B=(15,-19)`.
The distances from the origin to the new endpoints are
`d_A ~~22.8`
`d_B ~~24.2`

Explanation:

1. Reflection of a point with coordinates `(a_0,b_0)` relative to a line `x=6` (vertical line intersecting X-axis at coordinate `x=6`) will be horizontally shifted into a new X-coordinate obtained by adding to an X-coordinate of the axis of symmetry (`x=6`) the distance from it of the original X-coordinates (`6-a_0`).
   Y-coordinate remains the same in this transformation.
   So, new coordinates are:
   `(a_1,b_1) = (6+(6-a_0),b_0)=(12-a_0,b_0)`

2. Reflection of a point with coordinates `(a_1,b_1)` relative to a line `y=-1` (horizontal line intersecting Y-axis at coordinate `y=-1`) will be vertically shifted into a new Y-coordinate obtained by adding to an Y-coordinate of the axis of symmetry (`y=-1`) the distance from it of the original Y-coordinates (`-1-b_1`).
   X-coordinate remains the same in this transformation.
   So, new coordinates are:
   `(a_2,b_2) = (a_1,-1+(-1-b_1))=`
   `= (a_1,-2-b_1)=(12-a_0,-2-b_0)`

3. Dilation about a center point `(1,1)` by a factor of `2` will transform a point `(a_2,b_2)` into
   `(a_3,b_3) = (1+2(a_2-1),1+2(b_2-1)) =`
   `= (1+2(12-a_0-1),1+2(-2-b_0-1)) =`
   `= (23-2a_0, -5-2b_0)`

4. Using this formula for both ends of our original segment `AB`, where `A(1,2)` and `B=(4,7)`:
   4.1. `(a_0=1, b_0=2)`
   `rarr` `(a_3=23-2*1, b_3=-5-2*2) =`
   `= (21, -9)`
   4.2. `(a_0=4, b_0=7)`
   `rarr` `(a_3=23-2*4, b_3=-5-2*7) =`
   `= (15, -19)`

5. The distance of each end of a new segment from the origin are
   `d_A = sqrt((21)^2+(-9)^2) = sqrt(552) ~~22.8`
   `d_B = sqrt((15)^2+(-19)^2) = sqrt(586) ~~24.2`