# A line segment has endpoints at (3 ,5 ) and (2 ,6). 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?

Aug 4, 2016

$\left(3 , 5\right) \to \left(3 , 8\right) \text{ and } \left(2 , 6\right) \to \left(2.9\right)$

#### Explanation:

Since there are 3 transformations to be performed, name the endpoints A(3 ,5) and B(2 ,6) so that we can 'track' the points after each transformation.

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

a point (x ,y) → (-x ,y)

hence A(3 ,5) → A'(-3 ,5) and B(2 ,6) → B'(-2 ,6)

Second transformation Under a translation $\left(\begin{matrix}0 \\ 3\end{matrix}\right)$

a point (x ,y) → (x ,y+3)

hence A'(-3 ,5) → A''(-3 ,8) and B'(-2 ,6) → B''(-2 ,9)

Third transformation Under a reflection in the y-axis

a point (x ,y) → (-x ,y)

hence A''(-3 ,8) → A'''(3 ,8) and B''(-2 ,9) → B'''(2 ,9)

Hence after all 3 transformations.

$\left(3 , 5\right) \to \left(3 , 8\right) \text{ and } \left(2 , 6\right) \to \left(2 , 9\right)$