in the abcd parallelogram ad=10 units ab= 8 units ac=12 units ed= 4.5 units bd=

To find the length of BD in the ABCD parallelogram, we can use the properties of parallelograms and the given information.

Here's how you can find the length of BD:

Step 1: Understand the properties of a parallelogram:
A parallelogram is a quadrilateral with opposite sides that are parallel and congruent. In a parallelogram, opposite angles are also congruent. These properties provide a basis for applying various geometrical relationships in a parallelogram.

Step 2: Identify the relevant information:
In the given parallelogram ABCD, we are given the lengths of AD, AB, AC, and ED. We need to find BD. Let's write down the given lengths for reference:

AD = 10 units
AB = 8 units
AC = 12 units
ED = 4.5 units

Step 3: Use the properties of a parallelogram:
Since ABCD is a parallelogram, opposite sides are congruent. Therefore, we can conclude that AB = CD and AD = BC.

Step 4: Observe the given lengths:
We see that AD = 10 units and AB = 8 units. Using the property of opposite sides in a parallelogram, we can deduce that BC = AD = 10 units and CD = AB = 8 units.

Step 5: Apply the properties to find BD:
Using the property of opposite sides in a parallelogram, we can determine that BD = AB + CD.

BD = AB + CD
= 8 units + 8 units (since AB = CD)
= 16 units

Therefore, the length of BD in the ABCD parallelogram is 16 units.

using the law of cosines,

12^2 = 10^2 + 8^2 + 2*10*8 cosθ = 164+160cosθ
BD^2 = 10^2 + 8^2 - 2*8*10 cosθ
= 164 - 160(-20/160) = 184
BD = 13.56