A line segment has endpoints at (2 , 1) and (7 ,6). If the line segment is rotated about the origin by pi , translated horizontally by -3 , and reflected about the y-axis, what will the line segment's new endpoints be?

Jun 16, 2018

$\left(5 , - 1\right) \text{ and } \left(10 , - 6\right)$

Explanation:

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

$A = \left(2 , 1\right) \text{ and } B = \left(7 , 6\right)$

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

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

• " a point "(x,y)to(-x,-y)

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

$B \left(7 , 6\right) \to B ' \left(- 7 , - 6\right)$

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

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

• " a point "(x,y)to(x-3,y)

$A ' \left(- 2 , - 1\right) \to A ' ' \left(- 5 , - 1\right)$

$B ' \left(- 7 , - 6\right) \to B ' ' \left(- 10 , - 6\right)$

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

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

• " a point "(x,y)to(-x,y)

$A ' ' \left(- 5 , - 1\right) \to A ' ' ' \left(5 , - 1\right)$

$B ' ' \left(- 10 , - 6\right) \to B ' ' ' \left(10 , - 6\right)$

$\text{After all 3 transformations}$

$\left(2 , 1\right) \to \left(5 , - 1\right) \text{ and } \left(7 , 6\right) \to \left(10 , - 6\right)$