Question

A line segment has endpoints at `(6 ,5 )` and `(2 ,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

`(-4, 7); (0, 7)`

Explanation:

Use the Rotation, Translation and Reflection mattrix. But this a special case so you can do it in your head.
`vec(V') = M_(R(theta)) vec(V)`
`(-x_1, -y_1) = M_(R(theta))(x_1, y_1)`
`(-6, -7); (-2,-7)`

`vec(V'') = M_(T(2)) vec(V')`
`(-x_1 + 2, -y_1) = M_(T(2)) (-x_1, -y_1)`
`(-4, -7); (0,-7)`

`vec(V') = M_(RF(x) vec(V)`
`(-x_1 + 2, y_1) =M_(RF(x))(-x_1+2, -y_1)`
`(-4, 7); (0, 7)`