# A line segment has endpoints at (6 ,5 ) and (8 ,7 ). If the line segment is rotated about the origin by  pi , translated horizontally by  2 , and reflected about the x-axis, what will the line segment's new endpoints be?

Jul 18, 2018

color(violet)("After all 3 transformations " color(green)((7, 4)to(-4, -5)" and "(-7, -8)to(-7, 8)

#### Explanation:

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

$A \left(6 , 5\right) \text{ and } B \left(8 , 7\right)$

$\textcolor{m a r \infty n}{\text{transformation of rotation about the origin of }} \pi$

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

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

$\Rightarrow B \left(8 , 7\right) \to B ' \left(- 7 , - 8\right)$

$\textcolor{m a r \infty n}{\text{next transformation under a horizontal translation }} \left(\begin{matrix}2 \\ 0\end{matrix}\right)$

• " a point "(x,y)to(x + 2, y)

$\Rightarrow A ' \left(- 6 , - 5\right) \to A ' ' \left(- 4 , - 5\right)$

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

$\textcolor{m a r \infty n}{\text{last transformation under a reflection in the x-axis}}$

• " a point "(x,y)to(x,-y)

$\Rightarrow A ' ' \left(- 4 , - 5\right) \to A ' ' ' \left(- 4 , 5\right)$

$\Rightarrow B ' ' \left(- 7 , - 8\right) \to B ' ' ' \left(- 7 , 8\right)$

color(violet)("After all 3 transformations " color(green)((7, 4)to(-4, -5)" and "(-7, -8)to(-7, 8)