Parallelogram ABCD with (2,5), (4,9), (6,5), and (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?
The original D is ( 4,1)
When reflected across the x axis, 'D' is ( 4, -1)
D' and D" will lie on a line that will be perpendicular to y = x + 1
So....the slope of this line is - 1
And the equation of this line is
y = - ( x - 4) - 1
y = -x + 4 - 1
y = -x + 3
The intersection of these two lines will be the midpoint of D' and D"
So...setting the y's equal, we have
-x + 3 = x + 1 add x to both sides, subtract 1 from each side
2 = 2x divide both sides by 2
1 = x
And using either line to find the y coordinate of this intersection point, we have that
y = x + 1
y = 1 + 1
y = 2
So...the midpoint of D' and D" is ( 1, 2 )
So...using the midpoint formula with the points ( 4, -1) and (1,2), we can find D' as
(x + 4)/2 = 1 ( y + -1) / 2 = 2
Multiply both sides by 2
x + 4 = 2 y - 1 = 4
subtract 4 from both sides add 1 to each side
x = -2 y = 5
So... D" = (-2, 5)
Here's a graph that shows this :
https://www.desmos.com/calculator/rbi6tjxihp