To calculate the diagonal $BD$ in a quadrilateral, we need more information. Are you referring to a specific type of quadrilateral, like a rectangle or parallelogram? The methods can differ depending on the type.

Here are some general strategies:

1. Rectangle: For a rectangle, the diagonals are equal, and $BD$ can be found using the Pythagorean Theorem. If the sides $AB = a$ and $AD = b$, then: $BD = \sqrt{a^2 + b^2}$

2. Square: If the quadrilateral is a square, the diagonal $BD$ is: $BD = a\sqrt{2}$ where $a$ is the length of a side.

3. Parallelogram: For a general parallelogram, diagonal $BD$ is usually found using a combination of vector properties or the cosine rule, but if additional properties (like the height or angles) are known, other geometric relationships could be used.

4. Trapezium (Trapezoid): In an isosceles trapezoid, where the non-parallel sides are equal, diagonal $BD$ can be derived using symmetry and the Pythagorean theorem in specific triangles formed.

Please provide the type of quadrilateral and any known lengths, and I can give a more specific method for calculating diagonal $BD$ without the cosine rule.

Do you want more details or have any questions?

Here are five related questions:

1. How do you calculate the diagonal of a rhombus?
2. What is the formula for diagonals in a kite?
3. Can diagonal lengths of a parallelogram be equal?
4. How do you find diagonal lengths in an irregular quadrilateral?
5. How can vectors help in calculating diagonals?

Tip: The Pythagorean Theorem works in right-angled triangles, so you can often break complex shapes into triangles to find unknowns.