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

Nov 23, 2016

$\left(7 , 7\right)$ and $\left(8 , 2\right)$.

Explanation:

Moving a line segment is equivalent to moving its endpoints.

When a point $\left({x}_{0} , {y}_{0}\right)$ is rotated about the origin by $\frac{\pi}{2}$, the new point is always $\left({x}_{1} , {y}_{1}\right) = \left(\text{-} {y}_{0} , {x}_{0}\right)$. Think of it like this: if you're walking in the woods and holding a map so that "forward" is east, that's the same situation. The map's "right" (east) is your "forward", the map's "forward" (north) is your "left", etc.

So in a 1/4 turn counterclockwise, old right $\left({x}_{0}\right)$ becomes new up $\left({y}_{1}\right)$, and old up $\left({y}_{0}\right)$ becomes new left $\left(\text{-} {x}_{1}\right)$. This is the same as $\left({x}_{1} , {y}_{1}\right) = \left(\text{-} {y}_{0} , {x}_{0}\right)$.

So after rotating our points $\frac{\pi}{2}$ about the origin, the new points are:

$\left(\text{-} 4 , 7\right)$ and $\left(\text{-} 5 , 2\right)$.

Horizontal translations only affect your $x$-value, because they are a left-right ($x$-axis) shift, and not an up-down ($y$-axis) shift.

After translating both points horizontally by -$3$, our new points are:

$\left(\text{-} 7 , 7\right)$ and $\left(\text{-} 8 , 2\right)$.

Finally, reflecting a point about the $y$-axis simply means changing the sign of its $x$-coordinate. This reflection is a left-to-right flip, so the up-down $\left(y\right)$ location will not change. (If you flip a map over so that north and south stay "up" and "down", the map's "east" becomes left, and its "west" becomes right.)

After reflection about the $y$-axis, our final points will be:

$\left(7 , 7\right)$ and $\left(8 , 2\right)$.