# A line segment has endpoints at (3 ,7 ) and (4 ,9). If the line segment is rotated about the origin by (pi )/2 , translated vertically by 2 , and reflected about the y-axis, what will the line segment's new endpoints be?

Oct 2, 2016

$\left(3 , 7\right) \to \left(7 , 5\right) , \left(4 , 9\right) \to \left(9 , 6\right)$

#### Explanation:

Since there are 3 transformations to be performed, name the endpoints A(3 ,7) and B(4 ,9) so that we can follow the changes that occur to them.

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

a point (x ,y) → (-y ,x)

hence A(3 ,7) → A'(-7 ,3) and B(4 ,9) → B'(-9 ,4)

Second transformation Under a translation $\left(\begin{matrix}0 \\ 2\end{matrix}\right)$

a point (x ,y) → (x+0 ,y+2) → (x ,y+2)

hence A'(-7 ,3) → A''(-7 ,5) and B'(-9 ,4) → B''(-9 ,6)

Third transformation Under a reflection in the y-axis

a point (x ,y) → (-x ,y)

hence A''(-7 ,5) → A'''(7 ,5) and B''(-9 ,6) → B'''(9 ,6)

Thus after all 3 transformations.

$\left(3 , 7\right) \to \left(7 , 5\right) \text{ and } \left(4 , 9\right) \to \left(9 , 6\right)$