A line segment has endpoints at (7 ,5 ) and (2 ,6). If the line segment is rotated about the origin by (pi )/2 , translated vertically by 3 , and reflected about the x-axis, what will the line segment's new endpoints be?

Jan 6, 2018

$\left(- 5 , - 10\right) \text{ and } \left(- 6 , - 5\right)$

Explanation:

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

$A \left(7 , 5\right) \text{ and } B \left(2 , 6\right)$

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

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

• " a point "(x,y)to(-y,x)

$\Rightarrow A \left(7 , 5\right) \to A ' \left(- 5 , 7\right)$

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

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

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

• " a point "(x,y)to(x,y+3)

$\Rightarrow A ' \left(- 5 , 7\right) \to A ' ' \left(- 5 , 10\right)$

$\Rightarrow B ' \left(- 6 , 2\right) \to B ' ' \left(- 6 , 5\right)$

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

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

• " a point "(x,y)to(x,-y)

$\Rightarrow A ' ' \left(- 5 , 10\right) \to A ' ' ' \left(- 5 , - 10\right)$

$\Rightarrow B ' ' \left(- 6 , 5\right) \to B ' ' ' \left(- 6 , - 5\right)$

$\text{under all 3 transformations}$

$\left(7 , 5\right) \to \left(- 5 , - 10\right) \text{ and } \left(2 , 6\right) \to \left(- 6 , - 5\right)$