Determine if the following lengths make an acute, right or obtuse triangle. 10. 3, 4, 5.

To figure out what type of triangle is formed by the lengths 3, 4, and 5, we can use the Pythagorean theorem. For a right triangle, the square of the longest side must equal the sum of the squares of the other two sides. Here, 5 is the longest side, so we check: \(3^2 + 4^2 = 9 + 16 = 25\) and \(5^2 = 25\). Since they are equal, 3, 4, and 5 form a right triangle! If you're ever in doubt about triangle types, you can also use the basic properties of triangle inequalities. Remember: in an acute triangle, the square of the longest side is less than the sum of the squares of the other two sides; in an obtuse triangle, it's greater. This will help you classify any other triangles you encounter!