Given points A (1; 5), B (-2; 2), C (0; 0), D (3; 3) prove that a) ABCD is a parallelogram b) ABCD is a rectangle

Let’s start the solution by calculating the lengths of the opposite sides. It is known that opposite sides in a rectangle are equal to each other.

Let’s apply the formula for this:

AB = √ ((x1 – x2) ^ 2 – (y1 – y2) ^ 2).

Apply for each segment and obtain the lengths of the sides AB = 3√2; CD = 3√2. We got a pair of equal opposite sides.

And AD = BC = 2√2.

A quadrilateral whose opposite sides are equal is a parallelogram, that is, ABCD is a parallelogram.

Let’s find its diagonal AC and BD = √26. They are equal to each other, which means that the quadrilateral is a rectangle.