Find the area of a cyclic quadrilateral with sides, 4, 5, 6, and 7cm

Options

A. `800 cm^(2)`
B. ` sqrt""840 cm^(2)`
C. ` 840 cm ^(2)`
D. ` sqrt"" 420 cm ^(2)`

Answer

To find the area of a cyclic quadrilateral with sides 4 cm, 5 cm, 6 cm, and 7 cm, we can use Brahmagupta's formula, which states that the area \( A \) of a cyclic quadrilateral can be calculated using the semi-perimeter \( s \) and the lengths of the sides \( a, b, c, d \).

### Step-by-step Solution:

1. **Calculate the semi-perimeter \( s \)**:
\[ s = \frac{a + b + c + d}{2} \]
Here, \( a = 4 \, \text{cm}, b = 5 \, \text{cm}, c = 6 \, \text{cm}, d = 7 \, \text{cm} \).
\[ s = \frac{4 + 5 + 6 + 7}{2} = \frac{22}{2} = 11 \, \text{cm} \]

**Hint**: Remember that the semi-perimeter is half the sum of all sides.

2. **Apply Brahmagupta's formula**:
The area \( A \) of the cyclic quadrilateral can be calculated using the formula:
\[ A = \sqrt{s(s-a)(s-b)(s-c)(s-d)} \]

3. **Substitute the values into the formula**:
\[ A = \sqrt{11(11-4)(11-5)(11-6)(11-7)} \]
Calculate each term:
- \( s - a = 11 - 4 = 7 \)
- \( s - b = 11 - 5 = 6 \)
- \( s - c = 11 - 6 = 5 \)
- \( s - d = 11 - 7 = 4 \)

Thus, we have:
\[ A = \sqrt{11 \times 7 \times 6 \times 5 \times 4} \]

4. **Calculate the product inside the square root**:
First, calculate \( 7 \times 6 = 42 \).
Then, \( 42 \times 5 = 210 \).
Finally, \( 210 \times 4 = 840 \).
So, we have:
\[ A = \sqrt{11 \times 840} \]

5. **Calculate the final area**:
\[ A = \sqrt{9240} \]
Since \( 9240 \) can be simplified:
\[ A = \sqrt{840} \, \text{cm}^2 \]

### Final Answer:
The area of the cyclic quadrilateral is \( \sqrt{840} \, \text{cm}^2 \).

---