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\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}

Ask by Franklin Sullivan. in the United States
Feb 27,2025

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The side lengths 1.16 m, 21 m, and 39 m cannot form a triangle because the sum of the two smaller sides (1.16 m + 21 m = 22.16 m) is less than the largest side (39 m).

Solution

To determine if three side lengths can form a triangle, we must check that the sum of any two sides is greater than the third side. The three conditions are: 1) 1.16 m + 21 m > 39 m 2) 1.16 m + 39 m > 21 m 3) 21 m + 39 m > 1.16 m Let's check these one by one. 1. 1.16 m + 21 m = 22.16 m, which is NOT greater than 39 m. 2. 1.16 m + 39 m = 40.16 m, which is greater than 21 m. 3. 21 m + 39 m = 60 m, which is greater than 1.16 m. Since the first condition fails (22.16 m is less than 39 m), the side lengths do not satisfy the triangle inequality theorem and therefore cannot form a triangle.

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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!

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