# A line segment has endpoints at (3 ,7 ) and (4 ,5). If the line segment is rotated about the origin by (pi )/2 , translated vertically by -1 , and reflected about the y-axis, what will the line segment's new endpoints be?

Jul 14, 2017

$\left(7 , 2\right) \text{ and } \left(5 , 3\right)$

#### Explanation:

$\text{since there are 3 transformations to be performed}$

$\text{label the endpoints " A(3,7)" and } B \left(4 , 5\right)$

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

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

• " a point "(x,y)to(-y,x)

$\Rightarrow A \left(3 , 7\right) \to A ' \left(- 7 , 3\right) , B \left(4 , 5\right) \to B ' \left(- 5 , 4\right)$

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

$\text{under a translation of } \left(\begin{matrix}0 \\ - 1\end{matrix}\right)$

• " a point "(x,y)to(x,y-1)

$\Rightarrow A ' \left(- 7 , 3\right) \to A ' ' \left(- 7 , 2\right)$

$\Rightarrow B ' \left(- 5 , 4\right) \to B ' ' \left(- 5 , 3\right)$

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

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

• " a point "(x,y)to(-x,y)

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

$\Rightarrow B ' ' \left(- 5 , 3\right) \to B ' ' ' \left(5 , 3\right)$

$\text{after all 3 transformations}$

$\left(3 , 7\right) \to \left(7 , 2\right) \text{ and } \left(4 , 5\right) \to \left(5 , 3\right)$