Question 5

The four sequential sides of a quadrilateral have lengths $a=3.3 . b=6.8, c=7.5$, and $d=10.2$ (all measured in yards). The angle between the two smallest sides is $\alpha=92^{\circ}$.

What is the area of this figure?
\[
\text { area }=
\]
\[
\int \mathrm{yd}^{2}
\]
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Step 1 ：Given the four sequential sides of a quadrilateral with lengths \(a=3.3\), \(b=6.8\), \(c=7.5\), and \(d=10.2\) (all measured in yards). The angle between the two smallest sides is \(\alpha=92^{\circ}\).

Step 2 ：First, convert the angle to radians. \(\alpha = 1.6057029118347832\) radians.

Step 3 ：Next, calculate the length of the diagonal using the cosine rule. The diagonal is \(7.6613503647861085\) yards.

Step 4 ：Then, calculate the semi-perimeter. The semi-perimeter is \(s = 13.9\) yards.

Step 5 ：Finally, calculate the area using Brahmagupta's formula. The area is \(42.21559901268726\) square yards.

Step 6 ：Rounding to two decimal places, the area of the quadrilateral is \(\boxed{42.22}\) square yards.