Which set of dimensions describe an ambiguous triangle? Select one: \( \begin{array}{l}A=340, a=8 \mathrm{~cm}, b=10 \mathrm{~cm} \\ A=740, a=10 \mathrm{~cm}, b=8 \mathrm{~cm} \\ A=1040, b=12 \mathrm{~cm}, c=8 \mathrm{~cm} \\ a=12 \mathrm{in}, b=14 \mathrm{in}, c=20 \mathrm{in}\end{array} \)
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Beyond the Answer
An ambiguous triangle occurs when the given dimensions and angles do not guarantee a single triangle solution. One classic case is when two sides and an angle opposite one of the sides are known, leading to potentially two different triangles (or none at all). If we analyze the given options, the second one, with \( A=740, a=10 \mathrm{~cm}, b=8 \mathrm{~cm} \), leads to the possibility of two different triangle configurations based on the law of sines. This occurs because the angle \( A \) could lead to two different triangles based on the lengths of sides \( a \) and \( b \). The key to solving ambiguous triangle problems is understanding the conditions under which they arise, specifically the SSA (Side-Side-Angle) condition. When applying the Law of Sines in this scenario, ensure to check for all potential solutions, including the case where no triangle can be formed at all.
