A line segment has endpoints at (1 ,2 ) and (9 ,3 ). If the line segment is rotated about the origin by ( 3 pi)/2 , translated horizontally by  2 , and reflected about the x-axis, what will the line segment's new endpoints be?

Jul 19, 2018

color(brown)("After all 3 transformations " color(orange)((1, 2)to(4, 1)" and "(9, 3)to(5, 9)

Explanation:

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

$A \left(1 , 2\right) \text{ and } B \left(9 , 3\right)$

$\textcolor{\in \mathrm{di} g o}{\text{transformation of rotation about the origin of }} \frac{3 \pi}{2}$

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

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

$\Rightarrow B \left(9 , 3\right) \to B ' \left(3 , - 9\right)$

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

• " a point "(x,y)to(x + 2, y)

$\Rightarrow A ' \left(2 , - 1\right) \to A ' ' \left(4 , - 1\right)$

$\Rightarrow B ' \left(3 , - 9\right) \to B ' ' \left(5 , - 9\right)$

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

• " a point "(x,y)to(x, -y)

$\Rightarrow A ' ' \left(4 , - 1\right) \to A ' ' ' \left(4 , 1\right)$

$\Rightarrow B ' ' \left(5 , - 9\right) \to B ' ' ' \left(5 , 9\right)$

color(brown)("After all 3 transformations " color(orange)((1, 2)to(4, 1)" and "(9, 3)to(5, 9)

Jul 19, 2018

$\left(4 , 1\right) \text{ and } \left(5 , 9\right)$

Explanation:

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

$A = \left(1 , 2\right) \text{ and } B = \left(9 , 3\right)$

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

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

• " a point "(x,y)to(y,-x)

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

$B \left(9 , 3\right) \to B ' \left(3 , - 9\right)$

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

$\text{under a horizontal translation } \left(\begin{matrix}2 \\ 0\end{matrix}\right)$

• " a point "(x,y)to(x+2,y)

$A ' \left(2 , - 1\right) \to A ' ' \left(4 , - 1\right)$

$B ' \left(3 , - 9\right) \to B ' ' \left(5 , - 9\right)$

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

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

• " a point "(x,y)to(x,-y)

$A ' ' \left(4 , - 1\right) \to A ' ' ' \left(4 , 1\right)$

$B ' ' \left(5 , - 9\right) \to B ' ' ' \left(5 , 9\right)$

$\text{After all 3 transformations}$

$\left(1 , 2\right) \to \left(4 , 1\right) \text{ and } \left(9 , 3\right) \to \left(5 , 9\right)$