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 triangle according to the Law of Sines?

• A. $a=7$, $b=2$, $c=4$
• B. $a=2$, $b=2$, $c=5$
• C. $a=3$, $b=4$, $c=5$
• D. $a=6$, $b=6$, $c=12$

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 valid triangle, the sum of the angles must be \$180^\circ$, and each side length must satisfy the triangle inequality: the sum of any two sides must be greater than the third side.
3. Check each given set of side lengths against the triangle inequality: - For sides \(a\), \(b\), and \(c\), verify that \(a + b > c\), \(a + c > b\), and \(b + c > a\). - If any of these inequalities fail, the sides cannot form a triangle.
4. For the sets that satisfy the triangle inequality, consider the Law of Sines to see if the ratios of sides correspond to possible sine values of angles (which must be between 0 and 1). This helps confirm if the sides can correspond to valid angles.
5. Conclude which set(s) of side lengths can represent a triangle by confirming both the triangle inequality and the feasibility of the Law of Sines ratios, ensuring all angles are valid (between \$0^\circ\( and \)180^\circ$).