To calculate the area of triangle \( \triangle ABC \) given two sides and the included angle, we can use the formula:
\[ \text{Area} = \frac{1}{2} \cdot a \cdot b \cdot \sin(C) \]
Where:
- \( a \) and \( b \) are the lengths of the sides,
- \( C \) is the included angle.
From the problem, we have:
- \( a = 57 , \text{m} \)
- \( b = 65 , \text{m} \)
- \( \angle C = 86^\circ \)
We first need to calculate \( \sin(86^\circ) \). The sine of \( 86^\circ \) is approximately:
\[ \sin(86^\circ) \approx 0.998 \]
Now, substituting the values into the area formula:
\[ \text{Area} = \frac{1}{2} \cdot 57 \cdot 65 \cdot \sin(86^\circ) \] \[ \text{Area} \approx \frac{1}{2} \cdot 57 \cdot 65 \cdot 0.998 \] \[ \text{Area} \approx \frac{1}{2} \cdot 57 \cdot 65 \cdot 0.998 \approx \frac{1}{2} \cdot 57 \cdot 65 \cdot 0.998 \] \[ \text{Area} \approx 0.5 \cdot 57 \cdot 65 \cdot 0.998 \approx 1852.255 \text{ m}^2 \]
Now calculating the area:
\[ \text{Area} \approx 1852.255 , \text{m}^2 \]
Rounding to the nearest square meter:
\[ \text{Area} \approx 1852 , \text{m}^2 \]
Thus, the area of triangle \( \triangle ABC \) is approximately 1852 m².
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