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

Jul 2, 2018

$\left(- 5 , - 8\right) \text{ and } \left(- 7 , - 7\right)$

Explanation:

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

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

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

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

• " a point "(x,y)to(-y,x)

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

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

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

$\text{under a vertical translation } \left(\begin{matrix}0 \\ 2\end{matrix}\right)$

• " a point "(x,y)to(x,y+2)

$A ' \left(- 5 , 6\right) \to A ' ' \left(- 5 , 8\right)$

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

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

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

• " a point "(x,y)to(x,-y)

$A ' ' \left(- 5 , 8\right) \to A ' ' ' \left(- 5 , - 8\right)$

$B ' ' \left(- 7 , 7\right) \to B ' ' ' \left(- 7 , - 7\right)$

$\text{After all 3 transformations}$

$\left(6 , 5\right) \to \left(- 5 , - 8\right) \text{ and } \left(5 , 7\right) \to \left(- 7 , - 7\right)$