To find the length of the third side of the triangle, we can use the Law of Cosines. The Law of Cosines states:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]
where:
- \(a\) and \(b\) are the lengths of the two sides.
- \(C\) is the angle between those sides.
- \(c\) is the length of the side opposite to the angle \(C\).
Here, we have:
- \(a = 150 , \text{m}\)
- \(b = 200 , \text{m}\)
- \(C = 75^\circ\)
Now, substituting the values into the formula:
\[ c^2 = 150^2 + 200^2 - 2 \cdot 150 \cdot 200 \cdot \cos(75^\circ) \]
Calculating \(150^2\) and \(200^2\):
\[ 150^2 = 22500 \] \[ 200^2 = 40000 \]
Now adding these values together:
\[ c^2 = 22500 + 40000 - 2 \cdot 150 \cdot 200 \cdot \cos(75^\circ) \]
Next, we need to compute \(2 \cdot 150 \cdot 200\):
\[ 2 \cdot 150 \cdot 200 = 60000 \]
We also need to calculate \(\cos(75^\circ)\). The approximate value of \(\cos(75^\circ)\) is \(0.2588\).
Now substituting this value into the equation:
\[ c^2 = 62500 - 60000 \cdot 0.2588 \]
Calculating \(60000 \cdot 0.2588\):
\[ 60000 \cdot 0.2588 \approx 15528 \]
So the equation becomes:
\[ c^2 = 62500 - 15528 \] \[ c^2 \approx 46972 \]
Finally, taking the square root of both sides to find \(c\):
\[ c \approx \sqrt{46972} \approx 217 , \text{meters} \]
Thus, the length of the third side of the triangle is approximately 217 meters.
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