A line segment has endpoints at (2 ,0 ) and (1 , 3 ). If the line segment is rotated about the origin by  pi /2 , translated horizontally by  5 , and reflected about the y-axis, what will the line segment's new endpoints be?

Oct 21, 2016

$\left(2 , 0\right) \to \left(- 5 , 2\right) , \left(1 , 3\right) \to \left(- 2 , 1\right)$

Explanation:

There are 3 transformations to be performed here, so in order to follow the changes to the coordinates of the endpoints after each transformation label them as A(2 ,0) and B(1 ,3).

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

a point (x ,y) → (-y ,x)

hence A(2 ,0) → A'(0 ,2) and B(1 ,3) → B'(-3 ,1)

Second transformation Under a translation $\left(\begin{matrix}5 \\ 0\end{matrix}\right)$

a point (x ,y) → (x+5 ,y+0)

hence A'(0 ,2) → A''(5 ,2) and B'(-3 ,1) → B''(2 ,1)

Third transformation Under a reflection in the y-axis

a point (x ,y) → (-x ,y)

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

Thus after all 3 transformations.

$\left(2 , 0\right) \to \left(- 5 , 2\right) \text{ and } \left(1 , 3\right) \to \left(- 2 , 1\right)$