A line segment has endpoints at (7 ,2 ) and (3 ,4 ). If the line segment is rotated about the origin by (3 pi)/2 , translated vertically by 4 , and reflected about the y-axis, what will the line segment's new endpoints be?

May 11, 2018

$\left(- 2 , - 3\right) \text{ and } \left(- 4 , 1\right)$

Explanation:

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

$A = \left(7 , 2\right) \text{ and } B = \left(3 , 4\right)$

$\textcolor{b l u e}{\text{First transformation}}$
$\text{under a rotation about the origin of } \frac{3 \pi}{2}$

• " a point "(x,y)to(y,-x)

$\Rightarrow A \left(7 , 2\right) \to A ' \left(2 , - 7\right)$

$\Rightarrow B \left(3 , 4\right) \to B ' \left(4 , - 3\right)$

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

$\text{under a vertical translation } \left(\begin{matrix}0 \\ 4\end{matrix}\right)$

• " a point "(x,y)to(x,y+4)

$\Rightarrow A ' \left(2 , - 7\right) \to A ' ' \left(2 , - 3\right)$

$\Rightarrow B ' \left(4 , - 3\right) \to B ' ' \left(4 , 1\right)$

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

$\text{under a reflection about the y-axis}$

• " a point "(x,y)to(-x,y)

$\Rightarrow A ' ' \left(2 , - 3\right) \to A ' ' ' \left(- 2 , - 3\right)$

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

$\text{After all 3 transformations}$

$\left(7 , 2\right) \to \left(- 2 , - 3\right) \text{ and } \left(3 , 4\right) \to \left(- 4 , 1\right)$