Question

A line segment has endpoints at `(2 ,1 )` and `(3 ,5 )`. If the line segment is rotated about the origin by `( pi)/2`, translated vertically by `-8`, and reflected about the x-axis, what will the line segment's new endpoints be?

Answer 1

`P(1, -6) and Q(5 -5)`

Explanation:

Let `P(x, y)` then `P'(x',y') = R(theta) P(x,y)`
for `theta = pi/2`
`P'(x',y') = R(pi/2)P(x,y) = P'(-y,x)` this a special case of the generalized rotation operation.

Now `P(2,1) -> P'(-1,2)`
and `Q(3,5) -> Q'(-5,3)`
To translate by -8 simply subtract 8 to y part
`P'(-1, 2) -> P''(-1, -6)`
`Q'(-5,3) -> Q''(-5, -5)`
reflection about x axis will change the sign of point x
`P''(-1, -6)-> P'''(1, -6)`
`Q''(-5, -5)->Q'''(5 -5)`