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 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?
+128474 D = (4,1)
Reflecting this across the x axis gives D' = (4, -1)
Finding D" is a little tricky
The slope of the line through (4, -1) and perpendicular to y = x + 1 = -1
The equation of this line is
y = -1(x -4) - 1
y = -x + 4 - 1
y = -x + 3
The x coordinate of the intersection of this line and y = x + 1 can be found thusly
-x + 3 = x + 1
2x = 2
x = 1
And y = 2
And.....(1, 2) is the midpoint of (4, -1) and D"
So the find the x coordinate of D" we have
[ 4 + x] / 2 = 1
4 + x = 2
x = -2
And the y coordinate of D" is
[-1 + y ] /2 = 2
-1 + y = 4
y = 5
So
D" = ( -2, 5)
Here's the graph to show this : https://www.desmos.com/calculator/0nx7czfr1d
