Note that to show that a quadrilateral is a rectangle, it is sufficient to show that

(a) ABCD is a parallelogram, i.e., AC and BD bisect each other and,

(b) the diagonal AC and BD are equal

Let A(4, 8), B(0, 2), C(3, 0) and D(7, 6) are the vertices of a rectangle.

Coordinates of the midpoint of AC are

Coordinates of the midpoint of BD are

Thus, AC and BD have the same midpoint.

Hence, ABCD is a parallelogram

Now, check for the diagonals by using the distance formula

AC = √(3 – 4)² + (0 – 8)²

= √(-1)² + (-8)²

= √1 + 64

= √65 units

and

BD = √(7 - 0)² + (6 – 2)²

⇒ BD = √(7)² + (4)²

⇒ BD = √49 + 16

⇒ BD = √65 units

∴ AC = BD

Hence, ABCD is a rectangle.