Given a point A (0; 1) B (2; 5) C (4; 1) D (2; -3) Prove that ABCD is a rhombus.

Knowing the coordinates of two points, you can find the length of the segment connecting them using the formula:

l = √ ((x1 – x2) ^ 2 + (y1 – y2) ^ 2),

where x and y are the coordinates of the points.

Substituting the appropriate coordinates, you can find the lengths of the sides of the quadrilateral ABCD.

AB = √ ((0 – 2) ^ 2 + (1 – 5) ^ 2) = 2√5;

AD = √ ((0 – 2) ^ 2 + (1 + 3) ^ 2) = 2√5;

BC = √ ((2 – 4) ^ 2 + (5 – 1) ^ 2) = 2√5;

CD = √ ((4 – 2) ^ 2 + (1 + 3) ^ 2) = 2√5;

They are equal to each other, therefore, the investigated quadrangle is a rhombus.