# A line segment has endpoints at (2 , 2) and (5 , 4). If the line segment is rotated about the origin by (pi)/2 , translated vertically by 3, and reflected about the y-axis, what will the line segment's new endpoints be?

Mar 13, 2016

$\left(2 , 5\right)$ and $\left(4 , 8\right)$

#### Explanation:

Rotation through $\frac{\pi}{2}$ about the origin maps $\left(a , b\right)$ to $\left(- b , a\right)$

Vertical translation by $3$ maps $\left(a , b\right)$ to $\left(a , b + 3\right)$

Reflection about the $y$-axis maps $\left(a , b\right)$ to $\left(- a , b\right)$

Performing each of these in sequence:

$\left(a , b\right) \to \left(- b , a\right) \to \left(- b , a + 3\right) \to \left(b , a + 3\right)$

Hence:

$\left(2 , 2\right) \to \left(2 , 5\right)$

$\left(5 , 4\right) \to \left(4 , 8\right)$