# A line segment has endpoints at (1 ,2 ) and (3 ,8 ). 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?

May 16, 2018

$\left(2 , 2\right) \text{ and } \left(8 , 4\right)$

#### Explanation:

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

$A \left(1 , 2\right) \text{ and } B \left(3 , 8\right)$

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

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

• " a point "(x,y)to(-y,x)

$\Rightarrow A \left(1.2\right) \to A ' \left(- 2 , 1\right)$

rArrB(3,8)toB'*-8,3)

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

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

•" a point "(x,y)to(x,y+1)

$\Rightarrow A ' \left(- 2 , 1\right) \to A ' ' \left(- 2 , 2\right)$

$\Rightarrow B ' \left(- 8 , 3\right) \to B ' ' \left(- 8 , 4\right)$

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

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

• " a point "(x,y)to(-x,y)

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

$\Rightarrow B ' ' \left(- 8 , 4\right) \to B ' ' ' \left(8 , 4\right)$

$\text{After all 3 transformations}$

$\left(1 , 2\right) \to \left(2 , 2\right) \text{ and } \left(3 , 8\right) \to \left(8 , 4\right)$