Triangle LMN is an obtuse triangle with \( I=6, n=8.3 \), and \( m \angle M=132^{\circ} \). What is the area of \( \triangle L M N \) ? Round your answer to the nearest tenth. 10.2 square units 18.5 square units 24.9 square units 37.0 square units
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Bonus Knowledge
To find the area of triangle LMN, we can use the formula for the area of a triangle given two sides and the included angle: \[ \text{Area} = \frac{1}{2} \times a \times b \times \sin(C) \] In this case: - \( a = I = 6 \) - \( b = n = 8.3 \) - \( C = \angle M = 132^\circ \) Now, we need to calculate the sine of 132 degrees. Using a calculator or trigonometric tables, we find: \[ \sin(132^\circ) \approx 0.846 \] Now we can plug in the values: \[ \text{Area} = \frac{1}{2} \times 6 \times 8.3 \times 0.846 \] \[ \text{Area} = 3 \times 8.3 \times 0.846 \] \[ \text{Area} \approx 3 \times 7.0058 \approx 21.0174 \] Now, rounding it to the nearest tenth, we find: \[ \text{Area} \approx 21.0 \text{ square units} \] However, your answer options do not match this area calculation. Based on the computed area, it appears none of the options provided directly match this outcome, but among the options, 24.9 square units is the closest to the calculated area. Double-check all calculations to confirm the inputs and trigonometric outputs were accurate. The area of triangle LMN is approximately 21.0 square units, although it may warrant investigation if sticking to the provided answers!
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