Given three segments with lengths $a$, $b$, and $c$, which of the following sets of values could form a triangle according to the Law of Sines and the triangle inequality?

• A. $a=3$, $b=4$, $c=5$
• B. $a=5$, $b=5$, $c=10$
• C. $a=1$, $b=2$, $c=3$
• D. $a=2$, $b=3$, $c=6$

Answer: (A)

Verified step by step guidance

1. Recall the triangle inequality theorem, which states that for any three sides \(a\), \(b\), and \(c\) to form a triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. Mathematically, this means checking if all the following inequalities hold: \(a + b > c\), \(a + c > b\), and \(b + c > a\).
2. Apply the triangle inequality to each set of side lengths given in the problem. For example, for the set \(a=3\), \(b=4\), and \(c=5\), check if \$3 + 4 > 5\(, \)3 + 5 > 4\(, and \)4 + 5 > 3$.
3. Repeat the same process for the other sets: \(a=5\), \(b=5\), \(c=10\); \(a=1\), \(b=2\), \(c=3\); and \(a=2\), \(b=3\), \(c=6\). Verify whether each set satisfies all three inequalities.
4. Understand that if any one of these inequalities fails, the segments cannot form a triangle. This is because the sides would not be able to connect to enclose a space.
5. Once you identify which sets satisfy the triangle inequality, you can confirm that those sets could form a triangle. The Law of Sines can then be applied to find angles if needed, but the primary test for the possibility of a triangle is the triangle inequality.