\begin{tabular}{|l} Determine if the side lengths could form a triangl \\ \hline \( 1.16 \mathrm{~m}, 21 \mathrm{~m}, 39 \mathrm{~m} \) \\ 3\end{tabular}

To determine if the given side lengths can form a triangle, we can use the triangle inequality theorem. This theorem states that for any triangle with sides \(a\), \(b\), and \(c\), the following conditions must hold true: \(a + b > c\), \(a + c > b\), and \(b + c > a\). For the sides lengths \(1.16 \, \mathrm{m}\), \(21 \, \mathrm{m}\), and \(39 \, \mathrm{m}\): 1. \(1.16 + 21 = 22.16\) which is greater than \(39\) (fails). 2. \(1.16 + 39 = 40.16\) which is greater than \(21\) (true). 3. \(21 + 39 = 60\) which is greater than \(1.16\) (true). Since \(1.16 + 21\) is not greater than \(39\), the lengths \(1.16 \, \mathrm{m}\), \(21 \, \mathrm{m}\), and \(39 \, \mathrm{m}\) cannot form a triangle. --- For those who love trivia, did you know that the triangle inequality is used not just in geometry, but also in various fields like computer graphics and engineering? It helps ensure stability in structures and realistic modeling in animations! If you're looking to explore more about triangles, check out resources on Euclidean geometry for the foundational theories, or dive into trigonometry for applications in real-world scenarios, like navigation or architecture!