A line segment has endpoints at (7 ,5 ) and (8 ,2 ). If the line segment is rotated about the origin by  pi , translated horizontally by  - 1 , and reflected about the x-axis, what will the line segment's new endpoints be?

Jun 24, 2018

$\left(- 8 , 5\right) \text{ and } \left(- 9 , 2\right)$

Explanation:

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

$A = \left(7 , 5\right) \text{ and } B = \left(8 , 2\right)$

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

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

• " a point "(x,y)to(-x,-y)

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

$B \left(8 , 2\right) \to B ' \left(- 8 , - 2\right)$

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

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

• " a point "(x,y)to(x-1,y)

$A ' \left(- 7 , - 5\right) \to A ' ' \left(- 8 , - 5\right)$

$B ' \left(- 8 , - 2\right) \to B ' ' \left(- 9 , - 2\right)$

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

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

• " a point " (x,y)to(x,-y)

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

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

$\text{After all 3 transformations}$

$\left(7 , 5\right) \to \left(- 8 , 5\right) \text{ and } \left(8 , 2\right) \to \left(- 9 , 2\right)$