A line segment has endpoints at (2 ,3 ) and (8 ,9 ). If the line segment is rotated about the origin by ( pi)/2 , translated vertically by -8 , and reflected about the x-axis, what will the line segment's new endpoints be?

May 7, 2018

$\left(- 3 , 6\right) \text{ and } \left(- 9 , 0\right)$

Explanation:

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

$A \left(2 , 3\right) \text{ and } B \left(8 , 9\right)$

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

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

• " a point "(x,y)to(-y,x)

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

$\Rightarrow B \left(8 , 9\right) \to B ' \left(- 9 , 8\right)$

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

$\text{under a vertical translation } \left(\begin{matrix}0 \\ - 8\end{matrix}\right)$

• " a point "(x,y)to(x,y-8)

$\Rightarrow A ' \left(- 3 , 2\right) \to A ' ' \left(- 3 , - 6\right)$

$\Rightarrow B ' \left(- 9 , 8\right) \to B ' ' \left(- 9 , 0\right)$

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

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

• " a point "(x,y)to(x,-y)

$\Rightarrow A ' ' \left(- 3 , - 6\right) \to A ' ' ' \left(- 3 , 6\right)$

$\Rightarrow B ' ' \left(- 9 , 0\right) \to A ' ' ' \left(- 9 , 0\right)$

$\text{After all 3 transformations}$

$\left(2 , 3\right) \to \left(- 3 , 6\right) \text{ and } \left(8 , 9\right) \to \left(- 9 , 0\right)$