Given the points A(2,-2) B(1,1) C(1,4) D(x,5) find the missing coordinate if the segments AB and CD are parallel.

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First, we'll write the equations of the line MN and PQ.

y - yA = (yB - yA)(x - xA)/(xB - xA)

We'll substitute the coordinates of A and B:

y + 2 = (1+2)(x - 2)/(1 - 2)

y + 2 = 3(x-2)/-1

y = -3(x-2) - 2

y = -3x + 6 - 2

y = -3x + 4

The equation of the line AB is put in the points slope form, so that the slope of AB is mAB = -3

If AB and CD are parallel, then the values of their slopes are equal.

mAB=mCD = -3

mCD = (yD-yC)/(xD - xC)

mCD = (5-4)/(x-1)

mCD = 1/(x-1)

-3 = 1/(x-1)

-3*(x-1) = 1

-3x + 3 = 1

-3x = -2

x = 2/3

The missing coordinate is x = 2/3.

We have the points A(2,-2) B(1,1) C(1,4) and D(x,5) and we need to find x if AB and CD are parallel.

The slope of AB is (1+2)/(1-2) = 3/-1 = -3

The slope of CD is (5-4)/(x - 1) = 1/(x - 1)

For parallel lines the slope is the same

=> 1/(x - 1) = -3

=> 1 = -3x + 3

=> 3x = 2

=> x = 2/3

**The required value of x = 2/3**

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