Which of the following sets of three numbers can represent the side lengths of an obtuse triangle?

• A. $6,8,10$
• B. $5,5,8$
• C. $3,4,6$
• D. $7,8,15$

Answer: (C)

Verified step by step guidance

1. First, recall that for three side lengths to form a triangle, they must satisfy the triangle inequality: the sum of any two sides must be greater than the third side. Check this for each set to ensure they can form a triangle.
2. Next, to determine if a triangle is obtuse, use the converse of the Pythagorean theorem. Label the longest side as \(c\) and the other two sides as \(a\) and \(b\). Then compare \(c^2\) with \(a^2 + b^2\).
3. If \(c^2 > a^2 + b^2\), the triangle is obtuse. If \(c^2 = a^2 + b^2\), it is a right triangle. If \(c^2 < a^2 + b^2\), it is an acute triangle.
4. Apply this test to each set of side lengths: calculate \(c^2\), \(a^2\), and \(b^2\), then compare \(c^2\) with \(a^2 + b^2\) to determine the type of triangle.
5. Finally, identify which sets satisfy the obtuse condition (\(c^2 > a^2 + b^2\)) and confirm those sets can form a triangle based on the triangle inequality.