Question

the two opposite vertices of a square are(-1,2) and(3,2). find the co-ordinates of the other two vertices.

Answer 1

A(-1, 2).  B(a, b)... C(3,2)  D(c, d).

AC is parallel to x axis. So BD is parallel to y axis. So x coordinates of B and D are the same.  So  c = a.

AC = √(4²+0²) = 4.   So AB = BC = CD = DA = 2√2.

AB² = 8 = (a+1)² + (b-2)² 
       So  a² + 2a + b² - 4b = 3    -- (1)
BC² = 8 = (a-3)² + (b - 2)²
       So   a² - 6 a + b² - 4b = -5   --(2)

From (1) and (2),  a = 1. 
        So b - 2 = + 2     or,  b = 0 or 4

Since AC and BD are parallel to axes respectively, B and D are symmetrically above and below AC. The above two values of b are B and D.
        So   b = 0  and  d = 4.

So B = (1, 0)   and D (1, 4)