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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?

Dec 13, 2018

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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   Dec 13, 2018