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

Jan 12, 2018

$\left(3 , 5\right) \text{ and } \left(2 , 8\right)$

Explanation:

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

$\text{that is "A(2,3)" and } B \left(5 , 2\right)$

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

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

• " a point "(x,y)to(-y,x)

$\Rightarrow A \left(3 , 5\right) \to A ' \left(- 3 , 2\right)$

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

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

$\text{under a translation } \left(\begin{matrix}0 \\ 3\end{matrix}\right)$

• " a point "(x,y)to(x,y+3)

$\Rightarrow A ' \left(- 3 , 2\right) \to A ' ' \left(- 3 , 5\right)$

$\Rightarrow B ' \left(- 2 , 5\right) \to B ' ' \left(- 2 , 8\right)$

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

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

• " a point "(x,y)to(-x,y)

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

$\Rightarrow B ' ' \left(- 2 , 8\right) \to B ' ' ' \left(2 , 8\right)$

$\text{after all 3 transformations}$

$\left(2 , 3\right) \to \left(3 , 5\right) \text{ and } \left(5 , 2\right) \to \left(2 , 8\right)$