The two adjacent sides of a cyclic quadrilateral are `2` and `5` and the angle between them is `60^@`. If the area of the quadrilateral is `4sqrt(3)`, find the remaining two sides.

Options

A. `4` & `2`
B. `3` & `2`
C. `3` & `4`
D. `5` & `4`

Answer

To find the remaining two sides of the cyclic quadrilateral given the two adjacent sides (2 and 5) and the angle between them (60°), we can follow these steps:

### Step 1: Understand the Problem
We have a cyclic quadrilateral ABCD where:
- AB = 2
- BC = 5
- ∠ABC = 60°

### Step 2: Calculate the Area of Triangle ABC
The area of triangle ABC can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times AB \times BC \times \sin(\angle ABC) \]
Substituting the values:
\[ \text{Area} = \frac{1}{2} \times 2 \times 5 \times \sin(60^\circ) \]
Since \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\):
\[ \text{Area} = \frac{1}{2} \times 2 \times 5 \times \frac{\sqrt{3}}{2} = \frac{5\sqrt{3}}{2} \]

### Step 3: Total Area of Quadrilateral ABCD
The total area of the cyclic quadrilateral ABCD is given as \(4\sqrt{3}\). Therefore, the area of triangle ACD must be:
\[ \text{Area of ACD} = \text{Total Area} - \text{Area of ABC} \]
\[ \text{Area of ACD} = 4\sqrt{3} - \frac{5\sqrt{3}}{2} \]
To simplify, convert \(4\sqrt{3}\) into a fraction:
\[ 4\sqrt{3} = \frac{8\sqrt{3}}{2} \]
Thus,
\[ \text{Area of ACD} = \frac{8\sqrt{3}}{2} - \frac{5\sqrt{3}}{2} = \frac{3\sqrt{3}}{2} \]

### Step 4: Use the Area of Triangle ACD
Let the remaining sides be AC = x and AD = y. The area of triangle ACD can also be expressed as:
\[ \text{Area} = \frac{1}{2} \times AC \times AD \times \sin(\angle ACD) \]
Since angle ACD = 120° (as opposite angles in a cyclic quadrilateral sum to 180°):
\[ \sin(120^\circ) = \frac{\sqrt{3}}{2} \]
Thus,
\[ \frac{3\sqrt{3}}{2} = \frac{1}{2} \times x \times y \times \frac{\sqrt{3}}{2} \]
Multiplying both sides by 4:
\[ 6\sqrt{3} = x \times y \times \sqrt{3} \]
Dividing by \(\sqrt{3}\):
\[ 6 = x \times y \]

### Step 5: Use the Cosine Rule
Now, we can apply the cosine rule in triangle ABC to find the length of AC:
\[ AC^2 = AB^2 + BC^2 - 2 \cdot AB \cdot BC \cdot \cos(60^\circ) \]
Substituting the known values:
\[ AC^2 = 2^2 + 5^2 - 2 \cdot 2 \cdot 5 \cdot \frac{1}{2} \]
Calculating:
\[ AC^2 = 4 + 25 - 10 = 19 \]
Thus,
\[ AC = \sqrt{19} \]

### Step 6: Find the Remaining Side AD
Now we have:
\[ x = AC = \sqrt{19} \]
Using the equation \(x \cdot y = 6\):
\[ \sqrt{19} \cdot y = 6 \implies y = \frac{6}{\sqrt{19}} = \frac{6\sqrt{19}}{19} \]

### Final Result
The remaining two sides of the cyclic quadrilateral are:
- AC = \(\sqrt{19}\)
- AD = \(\frac{6\sqrt{19}}{19}\)