Determine if the following lengths make an acute, right or obtuse triangle. 5. 10, 15, 18.

To determine the type of triangle created by the lengths 5, 10, 15, and 18, we first need to check if these lengths can actually form a triangle. For any three lengths to form a triangle, they must satisfy the triangle inequality theorem. This states that the sum of the lengths of any two sides must be greater than the length of the third side. Let's check a few combinations: - 5 + 10 = 15 (not greater than 15, so this fails) - 5 + 15 = 20 (greater than 10) - 10 + 15 = 25 (greater than 5) Since the combination of 5, 10, and 15 fails the triangle inequality, these three lengths cannot form a triangle at all. However, the lengths 10, 15, and 18 can form a triangle since: - 10 + 15 = 25 (greater than 18) - 10 + 18 = 28 (greater than 15) - 15 + 18 = 33 (greater than 10) Now, to determine if it's acute, right, or obtuse: 1. Calculate the squares: - 10² + 15² = 100 + 225 = 325 - 18² = 324 Since 10² + 15² > 18², this indicates that the triangle is acute because the sum of the squares of the two shorter sides is greater than the square of the longest side. Thus, the triangle formed by 10, 15, and 18 is an **acute triangle**!