A student states that a triangle can be formed with side lengths 2 in, 3 in, and 6 in. Is the student correct? Why, or why not?
A. Yes, because [tex]$2+3\ \textless \ 6$[/tex]
B. Yes, because [tex]$6+2\ \textgreater \ 3$[/tex]
C. No, because [tex]$2+3\ \textless \ 6$[/tex]
D. No, because [tex]$6+2\ \textgreater \ 3$[/tex]
Answer (2)
Check if the sum of any two sides is greater than the third side.
Evaluate the condition 6"> 2 + 3 > 6 , which simplifies to 6"> 5 > 6 . This is false.
Since the triangle inequality is not satisfied, a triangle cannot be formed.
Conclude that the student is incorrect because 2 + 3 < 6 . No, because 2 + 3 < 6
Explanation
Understanding the Triangle Inequality Theorem To determine if a triangle can be formed with given side lengths, we need to apply the triangle inequality theorem. This theorem 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 Conditions Let's check the given side lengths: 2 in, 3 in, and 6 in.
We need to verify the following three conditions:
6"> 2 + 3 > 6
3"> 2 + 6 > 3
2"> 3 + 6 > 2
Evaluating the Conditions Let's evaluate each condition:
2 + 3 = 5 . Is 6"> 5 > 6 ? No, it is not.
2 + 6 = 8 . Is 3"> 8 > 3 ? Yes, it is.
3 + 6 = 9 . Is 2"> 9 > 2 ? Yes, it is.
Conclusion Since the first condition ( 6"> 2 + 3 > 6 ) is not met, the triangle inequality theorem is not satisfied. Therefore, a triangle cannot be formed with side lengths 2 in, 3 in, and 6 in.
Final Answer The student is incorrect because the sum of the two smaller sides (2 and 3) is not greater than the longest side (6).
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, when designing roofs or other triangular structures, architects must adhere to this theorem to ensure stability and prevent collapse.
A triangle cannot be formed with side lengths of 2 in, 3 in, and 6 in because the sum of the two shorter sides (2 in and 3 in) is not greater than the longest side (6 in). The correct answer is C: No, because 2 + 3 < 6 .
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