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

Apr 7, 2017

$\left(- 7 , 13\right) \text{ and } \left(- 5 , 6\right)$

#### Explanation:

Since there are 3 transformations to be performed, label the endpoints A(7 ,9) and B(5 ,2)

$\textcolor{b l u e}{\text{first transformation" " Under a rotation about origin of }} \pi$

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

$\text{Hence } A \left(7 , 9\right) \to A ' \left(- 7 , - 9\right)$

$\text{and } B \left(5 , 2\right) \to B ' \left(- 5 , - 2\right)$

$\textcolor{b l u e}{\text{Second transformation" " Under a translation }} \left(\begin{matrix}0 \\ - 4\end{matrix}\right)$

$\text{a point } \left(x , y\right) \to \left(x , y - 4\right)$

$\text{Hence } A ' \left(- 7 , - 9\right) \to A ' ' \left(- 7 , - 13\right)$

$\text{and } B ' \left(- 5 , - 2\right) \to B ' ' \left(- 5 , - 6\right)$

$\textcolor{b l u e}{\text{third transformation"" Under a reflection in the x-axis}}$

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

$\text{Hence } A ' ' \left(- 7 , - 13\right) \to A ' ' ' \left(- 7 , 13\right)$

$\text{and } B ' ' \left(- 5 , - 6\right) \to B ' ' ' \left(- 5 , 6\right)$

$\text{After all 3 transformations}$

$\left(7 , 9\right) \to \left(- 7 , 13\right) \text{ and } \left(5 , 2\right) \to \left(- 5 , 6\right)$