Final answer:
Option A (2 meters, 3 meters, 4 meters) is the correct set of side lengths that represent an acute triangle because it satisfies the triangle inequality theorem and the sum of the squares of the two shorter sides is greater than the square of the longest side.
Step-by-step explanation:
The question you've asked is about determining which set of given side lengths can form an acute triangle. An acute triangle is one where all three angles are less than 90 degrees. To verify if the given side lengths can form an acute triangle, we will use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's apply this to each option:
- A) 2 meters, 3 meters, 4 meters: 2 + 3 > 4, 3 + 4 > 2, and 2 + 4 > 3. This set satisfies the triangle inequality theorem and can form a triangle. To confirm it's an acute triangle, we also need to ensure that the sum of the squares of the two smaller sides is greater than the square of the longest side (22 + 32 > 42). This is indeed true (4 + 9 > 16), so option A represents an acute triangle.
- B) 2 meters, 3 meters, 5 meters: This does not satisfy the triangle inequality theorem since 2 + 3 is not greater than 5.
- C) 4 meters, 5 meters, 6 meters: This set satisfies the triangle inequality theorem, but when we check for it being an acute triangle (42 + 52 > 62), we find it's not true (16 + 25 is not greater than 36).
- D) 4 meters, 5 meters, 7 meters: This set similarly satisfies the triangle inequality theorem, but fails the check for an acute triangle since 42 + 52 is not greater than 72 (16 + 25 is not greater than 49).
Therefore, the correct answer is A) 2 meters, 3 meters, 4 meters, as this is the only option that satisfies all conditions to represent the side lengths of an acute triangle.