A line segment has endpoints at (4 ,5 ) and (2 ,3 ). If the line segment is rotated about the origin by ( 3 pi)/2 , translated horizontally by  - 1 , and reflected about the y-axis, what will the line segment's new endpoints be?

The new end points are $\left(- 2 , - 2\right)$ and $\left(- 4 , - 4\right)$

Explanation:

Considering the end point $\left(2 , 3\right)$ rotated $+ \frac{3 \pi}{2}$ about the origin $\left(0 , 0\right)$, it will end up exactly at $\left(3 , - 2\right)$. Translating that -1 horizontally will place it at $\left(2 , - 2\right)$ then reflecting about the y-axis will place it at $\left(- 2 , - 2\right)$ at the 3rd Quadrant.

Considering the end point $\left(4 , 5\right)$ rotated $+ \frac{3 \pi}{2}$ about the origin $\left(0 , 0\right)$, it will end up exactly at $\left(5 , - 4\right)$. Translating that -1 horizontally will place it at $\left(4 , - 4\right)$ then reflecting about the y-axis will place it at $\left(- 4 , - 4\right)$ at the 3rd Quadrant also.

Therefore the new end points are

$\left(- 2 , - 2\right)$ and $\left(- 4 , - 4\right)$

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