Given points A (1; 1), B (2; 3), C (0; 4), D (-1; 2). Show that the quadrilateral ABCD is a square.

We have points A (1; 1), B (2; 3), C (0; 4), D (-1; 2). Let’s prove that our figure is a square.

First, we find the values of the segments AB, BC, CD, AD:

AB = (1 + 4) ^ (1/2) = 5 ^ (1/2);

BC = (4 + 1) ^ (1/2) = 5 ^ (1/2);

CD = (1 + 4) ^ (1/2) = 5 ^ (1/2);

AD = (4 + 1) ^ (1/2) = 5 ^ (1/2);

ABCD is already a rhombus, since the sides are equal.

Now we are looking for the diagonals AC and BD of the quadrilateral:

AC = (1 + 3 ^ 2) ^ (1/2) = 10 ^ (1/2);

BD = (9 + 1) ^ (1/2) = 10 ^ (1/2);

The diagonals of the rhombus are equal – we have proved that the quadrilateral ABCD is a square.