Find the length of the longer diagonal of a parallelogram if the sides are 6 inches and 8 inches and the smaller angle is `60^(@)`

Options

A. 8
B. 11
C. 12
D. 7

Explanation

To find the length of the longer diagonal of a parallelogram with sides of 6 inches and 8 inches, and a smaller angle of 60 degrees, we can use the formula for the diagonal of a parallelogram.

### Step-by-step Solution:

1. **Identify the given values**:
- Side lengths: \( a = 8 \) inches, \( b = 6 \) inches
- Smaller angle: \( \theta = 60^\circ \)

2. **Use the formula for the diagonals of a parallelogram**:
The lengths of the diagonals \( d_1 \) and \( d_2 \) can be calculated using the formulas:
\[ d_1 = \sqrt{a^2 + b^2 + 2ab \cos(\theta)} \]
\[ d_2 = \sqrt{a^2 + b^2 - 2ab \cos(\theta)} \]
Here, \( d_1 \) is the longer diagonal and \( d_2 \) is the shorter diagonal.

3. **Calculate \( d_1 \)** (the longer diagonal):
- Substitute the values into the formula for \( d_1 \):
\[ d_1 = \sqrt{8^2 + 6^2 + 2 \cdot 8 \cdot 6 \cdot \cos(60^\circ)} \]
- Calculate \( \cos(60^\circ) = \frac{1}{2} \):
\[ d_1 = \sqrt{8^2 + 6^2 + 2 \cdot 8 \cdot 6 \cdot \frac{1}{2}} \]

4. **Simplify the expression**:
- Calculate \( 8^2 = 64 \) and \( 6^2 = 36 \):
\[ d_1 = \sqrt{64 + 36 + 2 \cdot 8 \cdot 6 \cdot \frac{1}{2}} \]
- The term \( 2 \cdot 8 \cdot 6 \cdot \frac{1}{2} = 48 \):
\[ d_1 = \sqrt{64 + 36 + 48} \]

5. **Combine the terms**:
\[ d_1 = \sqrt{148} \]

6. **Calculate the approximate value**:
- The square root of 148 can be simplified:
\[ d_1 \approx 12.17 \text{ inches} \]

### Final Answer:
The length of the longer diagonal of the parallelogram is approximately \( 12.17 \) inches.