To calculate the area of a quadrilateral given the side lengths and the angle between two adjacent sides, we can divide the quadrilateral into two triangles and calculate their areas separately. The two triangles are formed by drawing a diagonal.

Steps to Calculate the Area:

1. Break the quadrilateral into two triangles:

   • The two triangles are formed by the diagonal between the vertices of sides $5.7 \, \text{cm}$ and $12.8 \, \text{cm}$, as well as $7.2 \, \text{cm}$ and $9.4 \, \text{cm}$.

2. Calculate the area of the first triangle using the given angle:

   • The two smallest sides, $a = 5.7 \, \text{cm}$ and $b = 7.2 \, \text{cm}$, form an angle $\theta = 106^\circ$.
   • The area $A_1$ of this triangle can be calculated using the formula: $A_1 = \frac{1}{2}ab\sin\theta$ Substitute the values: $A_1 = \frac{1}{2} \times 5.7 \times 7.2 \times \sin(106^\circ)$

3. Calculate the length of the diagonal between the first two sides using the Law of Cosines:

   • The diagonal forms the third side $d$ of the triangle: $d^2 = a^2 + b^2 - 2ab\cos\theta$ Substituting the values: $d^2 = 5.7^2 + 7.2^2 - 2 \times 5.7 \times 7.2 \cos(106^\circ)$ Solve for $d$.

4. Calculate the area of the second triangle:

   • The second triangle is formed by the diagonal $d$ and the two remaining sides $c = 9.4 \, \text{cm}$ and $d_2 = 12.8 \, \text{cm}$.
   • To calculate its area, the Heron’s formula can be used: $\text{Semi-perimeter}, s = \frac{d + c + d_2}{2}$ $A_2 = \sqrt{s(s-d)(s-c)(s-d_2)}$

5. Add the two areas to find the total area: $A = A_1 + A_2$

Calculation:

Let’s proceed step-by-step.

Step 2: Calculate the area of the first triangle

$A_1 = \frac{1}{2} \times 5.7 \times 7.2 \times \sin(106^\circ)$ Using $\sin(106^\circ) \approx 0.961$: $A_1 \approx \frac{1}{2} \times 5.7 \times 7.2 \times 0.961 \approx 19.76 \, \text{cm}^2$

Step 3: Calculate the diagonal using the Law of Cosines

$d^2 = 5.7^2 + 7.2^2 - 2 \times 5.7 \times 7.2 \times \cos(106^\circ)$ Using $\cos(106^\circ) \approx -0.292$: $d^2 = 5.7^2 + 7.2^2 + 2 \times 5.7 \times 7.2 \times 0.292$ $d^2 \approx 32.49 + 51.84 + 24.02 \approx 108.35$ $d \approx \sqrt{108.35} \approx 10.41 \, \text{cm}$

Step 4: Calculate the area of the second triangle

$s = \frac{d + c + d_2}{2} = \frac{10.41 + 9.4 + 12.8}{2} \approx 16.31 \, \text{cm}$ $A_2 = \sqrt{s(s-d)(s-c)(s-d_2)}$ $A_2 = \sqrt{16.31(16.31 - 10.41)(16.31 - 9.4)(16.31 - 12.8)}$ $A_2 = \sqrt{16.31 \times 5.9 \times 6.91 \times 3.51} \approx \sqrt{2312.5} \approx 48.1 \, \text{cm}^2$

Step 5: Add the areas

$A = A_1 + A_2 \approx 19.76 + 48.1 \approx 67.86 \, \text{cm}^2$

Final Answer:

The area of the quadrilateral is approximately 67.9 cm².

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