Introduction:
In order to determine if a rectangle can have adjacent sides of lengths 2 cm and 4 cm, with diagonals bisecting each other at right angles, we need to analyze the properties of rectangles and their diagonals.

Properties of a Rectangle:
1. A rectangle is a quadrilateral with four right angles.
2. Opposite sides of a rectangle are equal in length.
3. The diagonals of a rectangle are equal in length and bisect each other.

Analysis:
Let's consider a rectangle with adjacent sides of lengths 2 cm and 4 cm.

1. Opposite Sides:
Since opposite sides of a rectangle are equal in length, the other pair of adjacent sides will also have lengths 2 cm and 4 cm.

2. Diagonals:
The diagonals of a rectangle are equal in length and bisect each other. In this case, the diagonals would intersect at their midpoints and form right angles.

Let's assume the shorter side of the rectangle is the width (2 cm) and the longer side is the length (4 cm).

Proof:
To prove that the diagonals of the rectangle bisect each other at right angles, we need to calculate the lengths of the diagonals and check if they are equal.

Using the Pythagorean theorem, the length of the diagonals can be calculated as follows:

Diagonal 1: sqrt((2 cm)^2 + (4 cm)^2) = sqrt(4 cm^2 + 16 cm^2) = sqrt(20 cm^2) = 2sqrt(5) cm
Diagonal 2: sqrt((4 cm)^2 + (2 cm)^2) = sqrt(16 cm^2 + 4 cm^2) = sqrt(20 cm^2) = 2sqrt(5) cm

Since both diagonals have the same length, we can conclude that they bisect each other.

However, the diagonals do not intersect at right angles. In this case, the diagonals would intersect at acute angles because the lengths of the sides do not satisfy the Pythagorean theorem to form right angles.

Conclusion:
In conclusion, a rectangle with adjacent sides of lengths 2 cm and 4 cm cannot have diagonals that bisect each other at right angles. The diagonals will intersect at acute angles instead.