A line segment has endpoints at (9 ,1 ) and (5 ,3). If the line segment is rotated about the origin by pi , translated vertically by -1 , and reflected about the x-axis, what will the line segment's new endpoints be?

Feb 27, 2018

$\left(- 9 , 2\right) \text{ and } \left(- 5 , 4\right)$

Explanation:

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

$A \left(9 , 1\right) \text{ and } B \left(5 , 3\right)$

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

$\text{under a rotation about the origin by } \pi$

• " a point "(x,y)to(-x,-y)

$\Rightarrow A \left(9 , 1\right) \to A ' \left(- 9 , - 1\right)$

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

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

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

• " a point "(x,y)to(x,y-1)

$\Rightarrow A ' \left(- 9 , - 1\right) \to A ' ' \left(- 9 , - 2\right)$

$\Rightarrow B ' \left(- 5 , - 3\right) \to B ' ' \left(- 5 , - 4\right)$

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

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

• " a point "(x,y)to(x,-y)

$\Rightarrow A ' ' \left(- 9 , - 2\right) \to A ' ' ' \left(- 9 , 2\right)$

$\Rightarrow B ' ' \left(- 5 , - 4\right) \to B ' ' ' \left(- 5 , 4\right)$

$\text{After all 3 transformations}$

$\left(9 , 1\right) \to \left(- 9 , 2\right) \text{ and } \left(5 , 3\right) \to \left(- 5 , 4\right)$