Could 14 inches, 13 inches, and 16 inches be the dimensions of a triangle? Explain your answer.
Answer (1)
Check if the sum of the two smaller sides is greater than the largest side.
Verify that 16"> 14 + 13 > 16 , 13"> 14 + 16 > 13 , and 14"> 13 + 16 > 14 .
Since all three inequalities hold true, the sides can form a triangle.
Final answer: T r u e
Explanation
Problem Analysis and Strategy Let's analyze the problem. We are given three side lengths: 14 inches, 13 inches, and 16 inches. We need to determine if these lengths can form a triangle. To do this, 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. We need to check all three possible combinations of sides.
Checking the Triangle Inequality Now, let's check the triangle inequality for all three combinations of sides:
14 + 13 > 16 27 > 16. This is true.
14 + 16 > 13 30 > 13. This is true.
13 + 16 > 14 29 > 14. This is true.
Since all three inequalities hold true, the given side lengths can form a triangle.
Conclusion Since all three triangle inequalities are satisfied, we can conclude that a triangle can be formed with sides of 14 inches, 13 inches, and 16 inches.
Examples
The triangle inequality is a fundamental concept in geometry and has practical applications in various fields. For example, in construction, when building a triangular structure like a roof truss, engineers must ensure that the lengths of the supports satisfy the triangle inequality. If the inequality is not satisfied, the structure will not be stable and could collapse. Similarly, in navigation, the shortest distance between two points is a straight line, which corresponds to one side of a triangle. The sum of the lengths of any other two sides (representing a detour) must be greater than the direct distance.
