Given triangle ABC with angles $A$, $B$, and $C$, and corresponding opposite sides $a$, $b$, and $c$, which of the following sets of side lengths could represent a possible triangle according to the Law of Sines?

• A. $a=5$, $b=5$, $c=11$
• B. $a=2$, $b=8$, $c=15$
• C. $a=7$, $b=10$, $c=12$
• D. $a=9$, $b=3$, $c=5$

Answer: (C)

Verified step by step guidance

1. Recall the Law of Sines, which states that for a triangle with sides \(a\), \(b\), and \(c\) opposite angles \(A\), \(B\), and \(C\) respectively, the following ratio holds: \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]
2. Understand that for any set of side lengths to form a valid triangle, they must satisfy the triangle inequality: the sum of any two sides must be greater than the third side. Check this first for each set of given side lengths.
3. For each set of side lengths, verify if the triangle inequality holds: - Check if \(a + b > c\) - Check if \(a + c > b\) - Check if \(b + c > a\)
4. If the triangle inequality is satisfied, use the Law of Sines to check if the ratios of sides to the sines of their opposite angles can be consistent. Since angles are unknown, consider if the side lengths could correspond to angles between 0° and 180° (excluding 0° and 180°).
5. Conclude that the set of side lengths that satisfy the triangle inequality and can correspond to valid angles (i.e., the Law of Sines ratios can be maintained with angles between 0° and 180°) represents a possible triangle.