The quadrilateral has sides equal to 12m, 20m, 8m, and 16.97m respectively. If the sum of the two opposite angles is equal to 225, find the area of the quadrilateral.
The correct answer and explanation is :
To find the area of a quadrilateral with given side lengths and the sum of two opposite angles, we will use Brahmagupta’s formula for the area of a cyclic quadrilateral.
Brahmagupta’s formula:
For a cyclic quadrilateral (a quadrilateral that can be inscribed in a circle), the area ( A ) can be calculated as:
[
A = \sqrt{(s – a)(s – b)(s – c)(s – d)}
]
Where:
- ( a, b, c, d ) are the lengths of the sides of the quadrilateral.
- ( s ) is the semi-perimeter of the quadrilateral, calculated as:
[
s = \frac{a + b + c + d}{2}
]
Given:
- Side lengths: ( a = 12 \, \text{m}, b = 20 \, \text{m}, c = 8 \, \text{m}, d = 16.97 \, \text{m} ).
- Sum of two opposite angles is 225°.
Since the sum of opposite angles is 225°, this implies that the quadrilateral is cyclic (it can be inscribed in a circle), allowing us to use Brahmagupta’s formula.
Step-by-Step Calculation:
- Find the semi-perimeter ( s ):
[
s = \frac{12 + 20 + 8 + 16.97}{2} = \frac{56.97}{2} = 28.485 \, \text{m}
]
- Apply Brahmagupta’s formula:
[
A = \sqrt{(28.485 – 12)(28.485 – 20)(28.485 – 8)(28.485 – 16.97)}
]
[
A = \sqrt{(16.485)(8.485)(20.485)(11.515)}
]
- Calculate the product inside the square root:
[
A = \sqrt{(16.485 \times 8.485 \times 20.485 \times 11.515)} \approx \sqrt{27316.768}
]
- Take the square root:
[
A \approx \sqrt{27316.768} \approx 165.3 \, \text{m}^2
]
Thus, the area of the quadrilateral is approximately 165.3 square meters.
Explanation:
In this problem, we use Brahmagupta’s formula to compute the area of the quadrilateral. The key information provided was that the quadrilateral is cyclic, which is implied by the fact that the sum of two opposite angles is 225°, a property of cyclic quadrilaterals. By calculating the semi-perimeter and using the formula, we determine the area. The result is approximately 165.3 square meters. This method is efficient for cyclic quadrilaterals, where opposite angles add up to 180° or a similar sum, and allows us to calculate the area directly without needing to divide the quadrilateral into triangles or use other methods like trigonometry.