To find the area of a quadrilateral when you are given the lengths of the sides, you can use Brahmagupta's formula if the quadrilateral is cyclic, but in this case, since we do not have information about the angles or if the quadrilateral is cyclic, we need to assume a way to calculate the area.
Given the sides:
- \(a = 7\) in
- \(b = 6\) in
- \(c = 10\) in
- \(d = 16\) in
We can calculate the semi-perimeter \(s\): \[ s = \frac{a + b + c + d}{2} = \frac{7 + 6 + 10 + 16}{2} = \frac{39}{2} = 19.5 \text{ in} \]
Next, using Brahmagupta's formula for the area \(A\) of a cyclic quadrilateral: \[ A = \sqrt{(s-a)(s-b)(s-c)(s-d)} \]
Calculating each term:
- \(s - a = 19.5 - 7 = 12.5\)
- \(s - b = 19.5 - 6 = 13.5\)
- \(s - c = 19.5 - 10 = 9.5\)
- \(s - d = 19.5 - 16 = 3.5\)
Now substituting back into the formula for area: \[ A = \sqrt{(12.5)(13.5)(9.5)(3.5)} \]
Calculating the product: \[ 12.5 \times 13.5 = 168.75 \] \[ 9.5 \times 3.5 = 33.25 \] \[ 168.75 \times 33.25 = 5615.625 \]
Finally, taking the square root: \[ A = \sqrt{5615.625} \approx 74.9 \text{ in.}^2 \]
Since this does not match any of the provided options directly, we can check to see if there’s a more straightforward approach to finding the area based on potential right triangle relationships or other configurations, but if we stick strictly to the provided side lengths and not assume they fit specific angles or configurations, we use this approximation.
None of the listed answers match exactly, but in a multiple-choice context, the closest response reflecting reasonable rounding would be not given here.
If you have additional context regarding the nature of the quadrilateral, such as confirming right angles or noting that it can be divided into triangles, please provide that and we can adjust the calculations appropriately.