Question

If (1, 1) and (1, 8) are the opposite vertices of a square, then find the co-ordinates of remaining two vertices.

Answer 1

Given: (1, 1) and (1, 8) are opposite vertices of a square.

Step wise calculation:

1. Midpoint center O of diagonal: 

`O = ((1 + 1)/2, (1 + 8)/2)`

= `(1, 9/2)`

2. Vector from O to A(1, 1): 

v = A – O

= `(0, 1 - 9/2)`

= `(0, -7/2)`

3. Rotate v by 90° to get the half-diagonal for the other pair: 

v_rot = (–y, x)

= `(7/2, 0)`

4. The remaining vertices are O ± v_rot:

O + v_rot = `(1 + 7/2, 9/2)`

= `(9/2, 9/2)`

O – v_rot = `(1 - 7/2, 9/2)`

= `(-5/2, 9/2)`

The other two vertices are `(9/2, 9/2)` and `(-5/2, 9/2)` equivalently (4.5, 4.5) and (–2.5, 4.5).