A line segment has endpoints at (4 ,2 ) and (3 ,1 ). 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?

Jun 28, 2018

$\left(2 , 5\right) \text{ and } \left(1 , 4\right)$

Explanation:

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

$A = \left(4 , 2\right) \text{ and } B = \left(3 , 1\right)$

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

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

• " a point "(x,y)to(-y,x)

$A \left(4 , 2\right) \to A ' \left(- 2 , 4\right)$

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

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

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

• " a point "(x,y)to(x,y+1)

$A ' \left(- 2 , 4\right) \to A ' ' \left(- 2 , 5\right)$

$B ' \left(- 1 , 3\right) \to B ' ' \left(- 1 , 4\right)$

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

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

• " a point "(x,y)to(-x,y)

$A ' ' \left(- 2 , 5\right) \to A ' ' ' \left(2 , 5\right)$

$B ' ' \left(- 1 , 4\right) \to B ' ' ' \left(1 , 4\right)$

$\text{After all 3 transformations }$

$\left(4 , 2\right) \to \left(2 , 5\right) \text{ and } \left(3 , 1\right) \to \left(1 , 4\right)$