Find the area of a square if the coordinates of its two vertices B (-1; 7) and D (5; -1) are given.

Let the square be designated ABCD, and given the coordinates of two points B (-1; 7) and D (5; -1). But judging by the points B and D, these are not neighboring points B and C, or A and B. So, BD is the diagonal, which has a relationship with the side of the square through the formula, and AB = BC = CD = AD.

CD ^ 2 + BC ^ 2 = BD ^ 2; 2 * ВС ^ 2 = ВD ^ 2. (one)

Determine BD from the data B (-1; 7) and D (5; -1): BD = √ [(- 1 -5) ^ 2 + (7 – (-1)] ^ 2 = √ [(- 6) ^ 2 + (7 + 1) ^ 2] = √ [36 + 64] = √100 = 10.

From formula (1) we determine BC ^ 2 = BD ^ 2/2 = 100/2 = 50.

And since the area is C = BC ^ 2 = 50.