A line segment has endpoints at (2 ,0 ) and (1 , 3 ). If the line segment is rotated about the origin by  pi /2 , translated horizontally by  1 , and reflected about the y-axis, what will the line segment's new endpoints be?

Mar 14, 2018

$\left(- 1 , 2\right) \text{ and } \left(2 , 1\right)$

Explanation:

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

$\text{Label the endpoints "A(2,0)" and } B \left(1 , 3\right)$

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

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

• " a point "(x,y)to(-y,x)

$\Rightarrow A \left(2 , 0\right) \to A ' \left(0 , 2\right)$

$\Rightarrow B \left(1 , 3\right) \to B ' \left(- 3 , 1\right)$

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

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

• " a point "(x,y)to(x+1,y)

$\Rightarrow A ' \left(0 , 2\right) \to A ' ' \left(1 , 2\right)$

$\Rightarrow B ' \left(- 3 , 1\right) \to B ' ' \left(- 2 , 1\right)$

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

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

• " a point "(x,y)to(-x,y)

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

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

$\text{After all 3 transformations}$

$\left(2 , 0\right) \to \left(- 1 , 2\right) \text{ and } \left(1 , 3\right) \to \left(2 , 1\right)$