Answer:
18.8 cm²
Explanation:
Sometimes, as here, when the problem is not carefully constructed, the answer you get depends on the method you choose for solving the problem.
Following directions
Using the formula ...
Area = (1/2)ab·sin(C)
we are given the values of "a" (BC=5.9 cm) and "b" (AC=7.2 cm), but we need to know the value of sin(C). The problem statement tells us to round this value to the nearest hundredth.
sin(C) = sin(118°) ≈ 0.882948 ≈ 0.88
Putting these values into the formula gives ...
Area = (1/2)(5.9 cm)(7.2 cm)(0.88) = 18.6912 cm² ≈ 18.7 cm² . . . rounded
You will observe that this answer does not match any offered choice.
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Rounding only at the End
The preferred method of working these problems is to keep the full precision the calculator offers until the final answer is achieved. Then appropriate rounding is applied. Using this solution method, we get ...
Area = (1/2)(5.9 cm)(7.2 cm)(0.882948) ≈ 18.7538 cm² ≈ 18.8 cm²
This answer matches the first choice.
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Using the 3 Side Lengths
Since the figure includes all three side lengths, we can compute a more precise value for angle C, or we can use Heron's formula for the area of the triangle. Each of these methods will give the same result.
From the Law of Cosines, the angle C is ...
C = arccos((a² +b² -c²)/(2ab)) = arccos(-38.79/84.96) ≈ 117.16585°
Note that this is almost 1 full degree less than the angle shown in the diagram. Then the area is ...
Area = (1/2)(5.9 cm)(7.2 cm)sin(117.16585°) ≈ 18.8970 cm² ≈ 18.9 cm²
This answer may be the most accurate yet, but does not match any offered choice.