A longer side of a parallelogram is 10 cm and the shorter is 6 cm. If the longer diagonal makes an angle `30^(@)` with the longer side, then the length of the longer diagonal (in cm) is

Options

A. `5 sqrt(3)+sqrt(11)`
B. `4 sqrt(3)+sqrt(11)`
C. `5 sqrt(3)+sqrt(13)`
D. `5 sqrt(3)-1`

Answer

To find the length of the longer diagonal of the parallelogram given the lengths of the sides and the angle, we can use the law of cosines. Let's solve the problem step by step.

### Step 1: Identify the given values
- Length of the longer side (AB) = 10 cm
- Length of the shorter side (BC) = 6 cm
- Angle between the longer side and the longer diagonal (AC) = 30 degrees

### Step 2: Set up the triangle
In triangle ABC:
- AB = 10 cm (longer side)
- BC = 6 cm (shorter side)
- Angle A = 30 degrees (angle between AB and AC)

### Step 3: Use the law of cosines
According to the law of cosines, the length of the diagonal AC can be calculated using the formula:
\[ AC^2 = AB^2 + BC^2 - 2 \cdot AB \cdot BC \cdot \cos(A) \]

### Step 4: Substitute the known values
Substituting the known values into the formula:
\[ AC^2 = 10^2 + 6^2 - 2 \cdot 10 \cdot 6 \cdot \cos(30^\circ) \]

### Step 5: Calculate the squares
Calculating the squares:
\[ AC^2 = 100 + 36 - 2 \cdot 10 \cdot 6 \cdot \cos(30^\circ) \]
\[ AC^2 = 100 + 36 - 120 \cdot \cos(30^\circ) \]

### Step 6: Find the value of cos(30 degrees)
The value of \(\cos(30^\circ)\) is \(\frac{\sqrt{3}}{2}\). Substituting this value:
\[ AC^2 = 100 + 36 - 120 \cdot \frac{\sqrt{3}}{2} \]
\[ AC^2 = 100 + 36 - 60\sqrt{3} \]

### Step 7: Simplify the equation
Now, simplifying the equation:
\[ AC^2 = 136 - 60\sqrt{3} \]

### Step 8: Calculate AC
To find AC, we take the square root:
\[ AC = \sqrt{136 - 60\sqrt{3}} \]

### Step 9: Approximate the value
Using a calculator, we can find the approximate value of \(AC\):
1. Calculate \(60\sqrt{3} \approx 60 \cdot 1.732 \approx 103.92\)
2. Then, \(136 - 103.92 \approx 32.08\)
3. Finally, \(AC \approx \sqrt{32.08} \approx 5.67\) cm (approximately).

### Final Answer
The length of the longer diagonal (AC) is approximately \(5.67\) cm.

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