A line segment has endpoints at (7 ,4 ) and (8 ,5 ). If the line segment is rotated about the origin by (3 pi)/2 , translated vertically by -2 , and reflected about the y-axis, what will the line segment's new endpoints be?

Feb 23, 2018

$\left(- 4 , - 9\right) \text{ and } \left(- 5 , - 10\right)$

Explanation:

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

$A \left(7 , 4\right) \text{ and } B \left(8 , 5\right)$

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

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

• " a point "(x,y)to(y,-x)

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

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

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

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

• " a point "(x,y)to(x,y-2)

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

$\Rightarrow B ' \left(5 , - 8\right) \to B ' ' \left(5 , - 10\right)$

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

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

• " a point "(x,y)to(-x,y)

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

$\Rightarrow B ' ' \left(5 , - 10\right) \to B ' ' ' \left(- 5 , - 10\right)$

$\text{After all 3 transformations}$

$\left(7 , 4\right) \to \left(- 4 , - 9\right) \text{ and } \left(8 , 5\right) \to \left(- 5 , - 10\right)$