Which of the following sets of numbers could represent the three sides of a triangle according to the Law of Sines ($\frac{a}{A}=\frac{b}{B}=\frac{c}{C}$)?

• A. $6$, $2$, $3$
• B. $1$, $2$, $3$
• C. $3$, $4$, $5$
• D. $2$, $2$, $5$

Answer: (C)

Verified step by step guidance

1. Recall that the Law of Sines states that for a triangle with sides \(a\), \(b\), and \(c\) opposite angles \(A\), \(B\), and \(C\) respectively, the following holds: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\).
2. To determine if a set of three numbers can represent the sides of a triangle, first check if they satisfy the triangle inequality: the sum of any two sides must be greater than the third side.
3. For each set of numbers, verify the triangle inequality: for example, check if \(a + b > c\), \(a + c > b\), and \(b + c > a\).
4. If the triangle inequality holds, then consider if there exist angles \(A\), \(B\), and \(C\) such that the Law of Sines can be satisfied. This is generally possible if the sides form a valid triangle.
5. Conclude that the set which satisfies the triangle inequality and can correspond to angles that satisfy the Law of Sines is the valid set of triangle sides.