Given triangle $ABC$, which of the following sets of side lengths could represent the sides of a triangle that satisfies the Law of Sines?

• A. $3$, $3$, $7$
• B. $7$, $10$, $12$
• C. $2$, $4$, $7$
• D. $5$, $9$, $15$

Answer: (B)

Verified step by step guidance

1. Recall that the Law of Sines states that for any triangle ABC, the ratio of the length of a side to the sine of its opposite angle is constant: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\).
2. Before applying the Law of Sines, verify if the given side lengths can form a valid triangle by checking the triangle inequality: the sum of any two sides must be greater than the third side.
3. Check each set of side lengths against the triangle inequality: for example, for sides 3, 3, and 7, check if \$3 + 3 > 7\(, \)3 + 7 > 3\(, and \)3 + 7 > 3$.
4. Eliminate any sets that do not satisfy the triangle inequality, as they cannot form a triangle and thus cannot satisfy the Law of Sines.
5. For the remaining valid sets, understand that the Law of Sines can be applied to find the angles, confirming that the ratios \(\frac{a}{\sin A}\), \(\frac{b}{\sin B}\), and \(\frac{c}{\sin C}\) are equal, which is consistent with the properties of a triangle.