To find the length of the third side of the triangle when two sides and the included angle are known, we can use the Law of Cosines. The Law of Cosines states:
\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]
Where:
- \( c \) is the length of the side opposite the angle \( C \),
- \( a \) and \( b \) are the lengths of the other two sides,
- \( C \) is the angle between sides \( a \) and \( b \).
In this case:
- \( a = 150 \) meters
- \( b = 200 \) meters
- \( C = 75^\circ \)
Now, substituting into the formula:
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Calculate \( a^2 \) and \( b^2 \): \[ a^2 = 150^2 = 22500 \] \[ b^2 = 200^2 = 40000 \]
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Calculate \( 2ab \cos(C) \):
- Calculate \( \cos(75^\circ) \): Using a calculator, \( \cos(75^\circ) \approx 0.2588 \). \[ 2ab \cos(75^\circ) = 2 \times 150 \times 200 \times 0.2588 \approx 2 \times 150 \times 200 \times 0.2588 \approx 15528 \]
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Now substitute into the Law of Cosines formula: \[ c^2 = 22500 + 40000 - 15528 \] \[ c^2 = 62500 - 15528 = 46972 \]
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Finally, find \( c \): \[ c = \sqrt{46972} \approx 216.7 \text{ meters} \]
Thus, the length of the third side of the triangle is approximately 216.7 meters.
The correct response is:
216.7 meters
1 answer