A line segment has endpoints at (0 ,0 ) and (1 , 3 ). If the line segment is rotated about the origin by  pi /2 , translated horizontally by  5 , and reflected about the y-axis, what will the line segment's new endpoints be?

Jan 22, 2018

$\left(- 5 , 0\right) \text{ and } \left(- 2 , 1\right)$

Explanation:

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

$A \left(0 , 0\right) \text{ and } B \left(1 , 3\right)$

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

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

• " a point "(x,y)to(-y,x)

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

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

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

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

• " a point "(x,y)to(x+5,y)

$\Rightarrow A ' \left(0 , 0\right) \to A ' ' \left(5 , 0\right)$

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

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

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

• " a point "(x,y)to(-x,y)

$\Rightarrow A ' ' \left(5 , 0\right) \to A ' ' ' \left(- 5 , 0\right)$

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

$\text{after all 3 transformations}$

$\left(0 , 0\right) \to \left(- 5 , 0\right) \text{ and } \left(1 , 3\right) \to \left(- 2 , 1\right)$