# A line segment has endpoints at (3 ,8 ) and (4 ,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?

Jan 2, 2017

$\left(8 , 6\right) \text{ and } \left(6 , 7\right)$

#### Explanation:

Since there are 3 transformations to be performed here, label the endpoints A(3 ,8) and B(4 ,6)

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

$\text{a point } \left(x , y\right) \to \left(- y , x\right)$

Hence A(3 ,8)→ A'(-8 ,3) and B(4 ,6) → B'(-6 ,4)

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

$\text{a point } \left(x , y\right) \to \left(x , y + 3\right)$

Hence A'(-8 ,3) → A''(-8 ,6) and B'(-6 ,4) → B''(-6 ,7)

Third transformation Under a reflection in the y-axis

$\text{a point } \left(x , y\right) \to \left(- x , y\right)$

Hence A''(-8 ,6) → A'''(8 ,6) and B''(-6 ,7) → B'''(6 ,7)

After all 3 transformations.

$\left(3 , 8\right) \to \left(8 , 6\right) \text{ and } \left(4 , 6\right) \to \left(6 , 7\right)$