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?

1 Answer
Nov 23, 2016

Answer:

#(7,7)# and #(8, 2)#.

Explanation:

Moving a line segment is equivalent to moving its endpoints.

When a point #(x_0,y_0)# is rotated about the origin by #pi/2#, the new point is always #(x_1,y_1)=("-"y_0,x_0)#. 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 #(x_0)# becomes new up #(y_1)#, and old up #(y_0)# becomes new left #("-"x_1)#. This is the same as #(x_1,y_1)=("-"y_0,x_0)#.

So after rotating our points #pi/2# about the origin, the new points are:

#("-"4, 7)# and #("-"5,2)#.

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:

#("-"7,7)# and #("-"8, 2)#.

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 #(y)# 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:

#(7,7)# and #(8, 2)#.