To find the area of a quadrilateral with given side lengths, we can use Brahmagupta's formula for a cyclic quadrilateral. However, we need to first determine if the quadrilateral can be cyclic (which simplifies our calculations) or if we need a different method based on the given dimensions.
Given your lengths are 4 cm, 9 cm, 6 cm, and 5 cm, we first find the semiperimeter \( s \) of the quadrilateral:
\[ s = \frac{a + b + c + d}{2} \] where \( a = 4 , \text{cm}, b = 9 , \text{cm}, c = 6 , \text{cm}, d = 5 , \text{cm} \).
Calculating the semiperimeter:
\[ s = \frac{4 + 9 + 6 + 5}{2} = \frac{24}{2} = 12 , \text{cm} \]
Now, we can use Brahmagupta's formula for the area \( A \):
\[ A = \sqrt{(s-a)(s-b)(s-c)(s-d)} \]
Substituting in our values:
\[ A = \sqrt{(12-4)(12-9)(12-6)(12-5)} = \sqrt{8 \cdot 3 \cdot 6 \cdot 7} \]
Calculating the inside of the square root:
\[ A = \sqrt{8 \cdot 3 \cdot 6 \cdot 7} = \sqrt{1008} \]
Now, calculating \( \sqrt{1008} \):
To simplify: \[ 1008 = 8 \times 126 = 8 \times 6 \times 21 = 48 \times 21 \]
So we find the square root: \[ \sqrt{1008} \approx 31.62 \]
This indicates that the area is approximately 32 cm², which does not correspond to the options given.
Another approach is dividing the quadrilateral into two triangles or using various methods based on additional information (like angles), but without that information, it is challenging to find the exact area.
If we have other relevant information such as diagonals or angles, we can find the exact area. Without additional information, the computations for area could lead to different results.
However, if they have explicitly stated choices such as 24 cm², 30 cm², 54 cm², and 60 cm² based on these sides, the most typically used formula or breakdown might be yielding those results.
Would you like me to use any other method or if you have more details on the quadrilateral?