Three vertices of square pqrs are p(–3, –2), q(–1, 3), and r(4, 1).
what are the coordinates of the fourth vertex?

I made a quick sketch of these three points on a scratch paper, and
I think this is the way to solve the problem.  Even though it seems
long and complicated, it's just doing things that you know how to do,
several times.

On paper, sketch and mark the three points.

You'll see that there are two sides of the square missing:
-- the side from 'P' to the missing point,
-- the side from 'R' to the missing point.

==> Find the slope of the side between 'Q' and 'R'.
Q=(-1, 3) . . . R=(4, 1)
Slope = (1-3) / (4 - -1) = -2/5
The side from 'P' to the missing point has the same slope,
and it goes through the point 'P' (-3, -2), so you can find its equation.
y = -2/5x + b
-2 = -2/5(-3) + b
-2 = 6/5  +  b
-2 - 6/5 = b = -10/5 - 6/5 = -16/5
y = -2/5x - 16/5

==> Find the slope of the side between 'P' and 'Q'.
The side from 'R' to the missing point has the same slope,
and it goes through the point 'R', so you can find its equation.

Maybe a bit shorter, you could say that this line is perpendicular to
the line between 'Q' and 'R' that you just found, so its slope is the
negative reciprocal of that one ... 5/2 instead of  -2/5s .

y = 5/2 x + b
This line goes through point 'R' (4, 1), so
1 = 5/2(4) + b
1 = 20/2 + b
1 = 10 + b
-9 = b
y = 5/2x - 9

Now you have the equations for the two lines that go through the
missing point.  When you solve this system of two equations, you'll
have the 'x'; and 'y' coordinates of the missing point, because that's
where they intersect.

y = -2/5x - 16/5  ,  and
y = 5/2x - 9

                                           -2/5 x - 16/5 = 5/2 x - 9
Multiply each side by 10 :      -4 x -  32  =  25 x - 90
Add  4x  to each side :                   - 32  =  29 x - 90
Add  90 to each side:                       58  =  29 x
Divide each side by  29 :          x = 2

y = 5/2 x - 9  =  5/2(2) - 9  =  5 - 9  =  -4

The missing corner is the point  (2, -4) .    

Like I said ... It seems long and complicated, but it's just a matter of
doing things that you know how to do, several times.  And trying to
not make any mistakes along the way (like I did about four times.)

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