A line segment has endpoints at (7 ,8 ) and (3 ,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?

Nov 6, 2016

to $A \left(7 , 8\right)$ corresponds $A ' ' ' \left(- 8 , - 9\right)$
to $B \left(3 , 5\right)$ corresponds B'''(-5;-5)

Explanation:

A rotation of $3 \frac{\pi}{2}$ anticlockwise is equivalent to a rotation of $\frac{\pi}{2}$ clockwise. In practice we we have to swap the abscissa with the ordinata payng attention to rigth sign to choose for each coordinate according to quadrant of destination after the rotation!

In our case, if $A \left(7 , 8\right)$ its corrispondent after rotation is $A ' \left(8 , - 7\right)$
and if $B \left(3 , 5\right)$ its corrispondent after rotation is $B ' \left(5 , - 3\right)$

If we translate vertically by -2, the rotated segment $A ' B '$, we have to lower the segment by 2 units. In practice we have to subtract 2 to the ordinates and we get
$A ' ' \left(8 , - 9\right)$ and #B''(5,-5)

The reflection about the y-axis implies to change the signs only to the abscissas
If we reflect about the y-axis, we get
$A ' ' ' \left(- 8 , - 9\right)$ and $B ' ' ' \left(- 5 , - 5\right)$