A line segment has endpoints at (0 ,2 ) and (3 ,5 ). If the line segment is rotated about the origin by ( 3 pi)/2 , translated vertically by  3 , and reflected about the x-axis, what will the line segment's new endpoints be?

Sep 12, 2016

$\left(2 , - 3\right) \text{ and } \left(5 , 0\right)$

Explanation:

Since there are 3 transformations to be performed here, name the endpoints A (0 ,2) and B (3 ,5) so that they may be 'tracked' after each transformation.

First transformation Under a rotation about the origin of $\frac{3 \pi}{2}$

a point (x ,y) → (y ,-x)

hence A(0 ,2) → A'(2 ,0) and B(3 ,5) → B'(5 ,-3)

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

a point (x ,y) → (x ,y+3)

hence A'(2 ,0) → A''(2 ,3) and B'(5 ,-3) → B''(5 ,0)

Third transformation Under a reflection in the x-axis

a point (x ,y) → (x ,-y)

hence A''(2 ,3) → A'''(2 ,-3) and B''(5 ,0) → B'''(5 ,0)

Thus after all 3 transformations:

$\left(0 , 2\right) \to \left(2 , - 3\right) \text{ and } \left(3 , 5\right) \to \left(5 , 0\right)$