One of diagonal of a parallelogram is 10 cm and an angle of the parallelogram is `pi//4`. If its height be 8 cm then area of the parallelogram :

Options

A. `112cm^(2)`
B. `88cm^(2)`
C. `92cm^(2)`
D. `104cm^(2)`

Answer

To find the area of the parallelogram given one diagonal, an angle, and the height, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values**:
- Diagonal (d) = 10 cm
- Angle (θ) = π/4 (which is 45 degrees)
- Height (h) = 8 cm

2. **Draw the Parallelogram**:
- Draw a parallelogram ABCD with diagonal AC.
- Mark the height from point B to line AC as DE, where DE = height = 8 cm.

3. **Analyze the Triangle Formed**:
- Triangle ABE is formed where:
- AB is the base,
- DE (or height) is perpendicular to AB,
- AE is the segment from A to the foot of the height (D).
- Since angle ABE is 45 degrees, triangle ABE is a right triangle.

4. **Use the Pythagorean Theorem**:
- In triangle ABE, we have:
- AE (height) = 8 cm
- AC (hypotenuse) = 10 cm
- Let EB be the base, then using the Pythagorean theorem:
\[ AE^2 + EB^2 = AC^2 \]
\[ 8^2 + EB^2 = 10^2 \]
\[ 64 + EB^2 = 100 \]
\[ EB^2 = 100 - 64 = 36 \]
\[ EB = 6 \text{ cm} \]

5. **Calculate the Base of the Parallelogram**:
- The base AB of the parallelogram is:
\[ AB = AE + EB = 8 + 6 = 14 \text{ cm} \]

6. **Calculate the Area of the Parallelogram**:
- The area (A) of a parallelogram is given by:
\[ A = \text{base} \times \text{height} \]
\[ A = AB \times h = 14 \times 8 = 112 \text{ cm}^2 \]

### Final Answer:
The area of the parallelogram is **112 cm²**.

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