Explanation:

To find the value of BD, we need to understand the properties of a parallelogram.

A parallelogram is a quadrilateral with opposite sides that are parallel and equal in length.

Properties of a parallelogram:

1. Opposite sides are parallel: In parallelogram ABCD, AB is parallel to CD and AD is parallel to BC.

2. Opposite sides are equal: In parallelogram ABCD, AB is equal to CD and AD is equal to BC.

3. Diagonals bisect each other: The diagonals AC and BD of parallelogram ABCD intersect at point O and bisect each other.

4. Diagonals divide each other in the ratio of the sides: The diagonals AC and BD divide each other in the ratio of the sides. This means that the ratio of AO to OC is equal to the ratio of BO to OD.

Using these properties, we can find the value of BD.

Solution:

Given that AB = a and BC = b.

Since AB is parallel to CD and AD is parallel to BC, we can say that AB is also equal to CD and AD is equal to BC. Therefore, CD = a and AD = b.

Since the diagonals bisect each other, AO = OC and BO = OD.

Using the property that the diagonals divide each other in the ratio of the sides, we can set up the following equation:

AO/OC = BO/OD

Since AO = OC and BO = OD, the equation becomes:

1 = 1

This equation is true, which means that the diagonals divide each other equally.

Therefore, BD is equal to half of the diagonal AC.

BD = 1/2 * AC

Since AC is equal to the sum of the sides AD and BC, we can substitute the values of AD and BC:

BD = 1/2 * (AD + BC)

BD = 1/2 * (b + b)

BD = 1/2 * 2b

BD = b

Therefore, the value of BD is equal to b.