# A line segment has endpoints at (7 ,4 ) and (5 ,9). If the line segment is rotated about the origin by (pi )/2 , translated vertically by 5 , and reflected about the y-axis, what will the line segment's new endpoints be?

Jun 7, 2017

$\left(- 4 , - 2\right) \text{ and } \left(- 9 , 0\right)$

#### Explanation:

Since there are 3 transformations to be performed, label the endpoints.
$A \left(7 , 4\right) \text{ and } B \left(5 , 9\right)$

$\textcolor{b l u e}{\text{First transformation }}$

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

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

$\Rightarrow A \left(7 , 4\right) \to A ' \left(- 4 , 7\right) \text{ and } B \left(5 , 9\right) \to B ' \left(- 9 , 5\right)$

$\textcolor{b l u e}{\text{Second transformation}}$

$\text{Under a translation of } \left(\begin{matrix}0 \\ - 5\end{matrix}\right)$

$\text{a point } \left(x , y\right) \to \left(x , y - 5\right)$

$\Rightarrow A ' \left(- 4 , 7\right) \to A ' ' \left(- 4 , 2\right)$

$\text{and } B ' \left(- 9 , 5\right) \to B ' ' \left(- 9 , 0\right)$

$\textcolor{b l u e}{\text{Third transformation}}$

$\text{Under a reflection in the y-axis}$

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

$\Rightarrow A ' ' \left(- 4 , 2\right) \to A ' ' ' \left(- 4 , - 2\right)$

$\text{and } B ' ' \left(- 9 , 0\right) \to B ' ' ' \left(- 9 , 0\right)$

$\text{After all 3 transformations}$

$\left(7 , 4\right) \to \left(- 4 , - 2\right) \text{ and } \left(5 , 9\right) \to \left(- 9 , 0\right)$