To find the area of the parallelogram given one side and the lengths of its diagonals, we can follow these steps:
### Step 1: Understand the properties of the parallelogram
We know that the diagonals of a parallelogram bisect each other. This means that if we label the diagonals as \( AC \) and \( BD \), they will intersect at point \( O \).
### Step 2: Label the given information
- Let \( AB = CD = 10 \, \text{cm} \) (one side of the parallelogram).
- Let \( BD = 12 \, \text{cm} \) (one diagonal).
- Let \( AC = 16 \, \text{cm} \) (the other diagonal).
### Step 3: Find the lengths of the segments formed by the diagonals
Since the diagonals bisect each other:
- \( BO = OD = \frac{BD}{2} = \frac{12}{2} = 6 \, \text{cm} \)
- \( AO = OC = \frac{AC}{2} = \frac{16}{2} = 8 \, \text{cm} \)
### Step 4: Calculate the area of triangle \( COD \)
The area of triangle \( COD \) can be calculated using the formula:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base \( CD = 10 \, \text{cm} \) and the height \( H \) is unknown. So, we can express the area as:
\[
\text{Area}_{COD} = \frac{1}{2} \times 10 \times H = 5H \, \text{cm}^2
\]
### Step 5: Use Heron's formula to find the area of triangle \( COD \)
Using Heron's formula, we first calculate the semi-perimeter \( s \):
\[
s = \frac{AO + BO + CD}{2} = \frac{8 + 6 + 10}{2} = 12 \, \text{cm}
\]
Now, we can apply Heron's formula:
\[
\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}
\]
Where \( a = 10 \, \text{cm} \), \( b = 6 \, \text{cm} \), and \( c = 8 \, \text{cm} \):
\[
\text{Area}_{COD} = \sqrt{12 \times (12 - 10) \times (12 - 6) \times (12 - 8)}
\]
Calculating this:
\[
= \sqrt{12 \times 2 \times 6 \times 4} = \sqrt{576} = 24 \, \text{cm}^2
\]
### Step 6: Set the areas equal to find height \( H \)
From Step 4 and Step 5, we have:
\[
5H = 24
\]
Solving for \( H \):
\[
H = \frac{24}{5} = 4.8 \, \text{cm}
\]
### Step 7: Calculate the height of the parallelogram
Since the height of the triangle \( COD \) is half the height of the parallelogram, the total height \( h \) of the parallelogram is:
\[
h = 2H = 2 \times 4.8 = 9.6 \, \text{cm}
\]
### Step 8: Calculate the area of the parallelogram
The area \( A \) of the parallelogram is given by:
\[
A = \text{base} \times \text{height} = 10 \times 9.6 = 96 \, \text{cm}^2
\]
### Final Answer
The area of the parallelogram is \( 96 \, \text{cm}^2 \).
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