# A line segment has endpoints at (4 ,7 ) and (5 ,6). If the line segment is rotated about the origin by pi , translated vertically by -2 , and reflected about the x-axis, what will the line segment's new endpoints be?

Jul 18, 2018

color(chocolate)("After all 3 transformations " color(gray)((4, 7)to(-4, 9)" and "(5, 6)to(-5, 8)

#### Explanation:

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

$A \left(4 , 7\right) \text{ and } B \left(5 , 6\right)$

$\textcolor{\in \mathrm{di} g o}{\text{transformation of rotation about the origin of }} \left(\pi\right)$

$\text{ a point } \left(x , y\right) \to \left(- x , - y\right)$

$\Rightarrow A \left(4 , 7\right) \to A ' \left(- 4 , - 7\right)$

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

$\textcolor{\in \mathrm{di} g o}{\text{next transformation under a vertical translation }} \left(\begin{matrix}0 \\ - 2\end{matrix}\right)$

• " a point "(x,y)to(x, y - 2)

$\Rightarrow A ' \left(- 4 , - 7\right) \to A ' ' \left(- 4 , - 9\right)$

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

$\textcolor{\in \mathrm{di} g o}{\text{last transformation under a reflection in the x -axis}}$

• " a point "(x,y)to(x, -y)

$\Rightarrow A ' ' \left(- 4 , - 9\right) \to A ' ' ' \left(- 4 , 9\right)$

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

color(chocolate)("After all 3 transformations " color(gray)((4, 7)to(-4, 9)" and "(5, 6)to(-5, 8)