# A line segment has endpoints at (2 , 2) and (5 , 4). If the line segment is rotated about the origin by (pi)/2 , translated horizontally by 1, and reflected about the y-axis, what will the line segment's new endpoints be?

Jul 28, 2016

$\left(2 , 2\right) \to \left(1 , 2\right) \text{ and } \left(5 , 4\right) \to \left(3 , 5\right)$

#### Explanation:

Since there are 3 transformations, name the endpoints A(2 , 2) and B(5 ,4) so we can 'track' their position after each transformation.

First transformation Under a rotation about the origin by $\frac{\pi}{2}$

a point (x ,y) → (-y ,x)

hence A(2 ,2) → A'(-2 ,2) and B(5 ,4) → B'(-4 ,5)

Second transformation Under a translation $\left(\begin{matrix}1 \\ 0\end{matrix}\right)$

a point (x ,y) → (x+1 ,y)

hence A'(-2 ,2) → A''(-1 ,2) and B'(-4 ,5) → B''(-3 ,5)

Third transformation Under a reflection in the y-axis

a point (x ,y) → (-x ,y)

hence A''(-1 ,2) → A'''(1 ,2) and B''(-3 ,5) → B'''(3 ,5)

Thus $\left(2 , 2\right) \to \left(1 , 2\right) \text{ and } \left(5 , 4\right) \to \left(3 , 5\right)$