Decide whether ABCD with vertices A (3,8) B (6,5) C (5,4) , and D (2,7) is a rectangle a rhombus or a square

Given:

The vertices of the quadrilateral ABCD are A(3,8), B(6,5), C(5,4), and D(2,7).

To find:

Whether the given quadrilateral is a rectangle, a rhombus or a square.

Solution:

Distance formula: The distance between two points is

Using distance formula, the side lengths are:

Similarly,

The length  of diagonals are:

From the above calculation, we conclude that the given quadrilateral has two pairs of congruent opposite sides and equal diagonals.

Opposite sides of a rectangle are equal and its diagonals are also equal.

Therefore, the given quadrilateral ABCD is a rectangle.