Question

PLEASE ANSWER CORRECTLY, DON'T JUST STEAL THE POINTS.

Parallelogram ABCD with A(2,5), B(4,9), C(6,5), and D(4,1) is reflected across the x-axis to A'B'C'D' and then A'B'C'D' is reflected across the line y=x+1 to A''B''C''D''. This is done such that D' is the image of D, and D'' is the image of D'. What is the ordered pair of D'' in the coordinate plane?

Answer 1

Since you're only asked the ordered pair of D'', it's much easier just to plot and reflect point D twice than to do that for all four points!

Remember that reflecting points is like putting a mirror at the line of reflection or flipping that point over at that line. The reflected point should be the same distance from the line of reflection as the original point.

1) Reflect D over the x-axis to get D'.
D is at (4,1). Draw a line that is perpendicular to the line of reflection and goes through D. D is as far from the line of reflection as D' should be on its other side (both are on that perpendicular line). Since D is 1 unit above the x-axis, that means D' is 1 unit below at (4, -1). See picture 1.

2) Reflect D' over y=x+1 to get D''.
D' is at (4, -1). Draw y=x+1 and the line perpendicular to it going through D''. D'' is the same distance from the line of reflection on the other side. See picture 2. D'' is at (-2, 5).

Answer: D'' is at (-2, 5)