A line segment has endpoints at (1 ,2 ) and (9 ,4 ). If the line segment is rotated about the origin by ( 3 pi)/2 , translated vertically by  3 , 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(maroon)((1, 2)to(-2, -2)" and "(9, 4)to(4, 6)

Explanation:

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

$A \left(1 , 2\right) \text{ and } B \left(9 , 4\right)$

$\textcolor{p u r p \le}{\text{transformation of rotation about the origin of }} \frac{3 \pi}{2}$

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

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

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

$\textcolor{p u r p \le}{\text{next transformation under a vertical translation }} \left(\begin{matrix}0 \\ 3\end{matrix}\right)$

• " a point "(x,y)to(x, y + 3)

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

$\Rightarrow B ' \left(4 , - 9\right) \to B ' ' \left(4 , - 6\right)$

$\textcolor{p u r p \le}{\text{last transformation under a reflection in the x -axis}}$

• " a point "(x,y)to(x, -y)

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

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

color(chocolate)("After all 3 transformations " color(maroon)((1, 2)to(-2, -2)" and "(9, 4)to(4, 6)

Jul 18, 2018

$\left(2 , - 2\right) \text{ and } \left(4 , 6\right)$

Explanation:

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

$A = \left(1 , 2\right) \text{ and } B = \left(9 , 4\right)$

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

$\text{under a rotation about the origin of } \frac{3 \pi}{2}$

• " a point "(x,y)to(y,-x)

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

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

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

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

$\text{a point } \left(x , y\right) \to \left(x , y + 3\right)$

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

$B ' \left(4 , - 9\right) \to B ' ' \left(4 , - 6\right)$

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

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

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

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

$B ' ' \left(4 , - 6\right) \to B ' ' ' \left(4 , 6\right)$

$\text{After all 3 transformations}$

$\left(1 , 2\right) \to \left(2 , - 2\right) \text{ and } \left(9 , 4\right) \to \left(4 , 6\right)$