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 first two sides is greater than the third side: 16"> 14 + 13 > 16 .
Check if the sum of the first and third sides is greater than the second side: 13"> 14 + 16 > 13 .
Check if the sum of the second and third sides is greater than the first side: 14"> 13 + 16 > 14 .
Since all three inequalities hold, the side lengths can form a triangle: True .
Explanation
Problem Analysis and Strategy Let's analyze the problem. We are given three lengths, 14 inches, 13 inches, and 16 inches, and 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.
Checking the Triangle Inequality Now, let's check if the triangle inequality theorem holds for all three combinations of sides:
14 inches + 13 inches > 16 inches
14 inches + 16 inches > 13 inches
13 inches + 16 inches > 14 inches
Evaluating the Inequalities Let's evaluate each inequality:
14 + 13 = 27 . Since 16"> 27 > 16 , the first inequality holds.
14 + 16 = 30 . Since 13"> 30 > 13 , the second inequality holds.
13 + 16 = 29 . Since 14"> 29 > 14 , the third inequality holds.
Conclusion Since all three inequalities hold true, the given side lengths can form a triangle.
Examples
The triangle inequality theorem is useful in construction and engineering. For example, when building a bridge, engineers need to ensure that the lengths of the supporting beams satisfy the triangle inequality to guarantee the structural integrity of the bridge. Similarly, in architecture, the theorem helps in designing stable triangular structures.
