Question
Can the side lengths of a triangle be 3, 6 and 9? (1 point)
Answers
GPT-4o mini
To determine if lengths 3, 6, and 9 can form a triangle, we can apply the triangle inequality theorem, which states that for any triangle with sides \(a\), \(b\), and \(c\), the following conditions must hold:
1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)
Let’s apply this to the side lengths 3, 6, and 9. We can assign:
- \(a = 3\)
- \(b = 6\)
- \(c = 9\)
Now we check the triangle inequalities:
1. \(3 + 6 > 9\) → \(9 > 9\) (not true)
2. \(3 + 9 > 6\) → \(12 > 6\) (true)
3. \(6 + 9 > 3\) → \(15 > 3\) (true)
Since the first condition is not satisfied (as \(9\) is not greater than \(9\)), the lengths 3, 6, and 9 cannot form a triangle. Therefore, the answer is **No**.
1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)
Let’s apply this to the side lengths 3, 6, and 9. We can assign:
- \(a = 3\)
- \(b = 6\)
- \(c = 9\)
Now we check the triangle inequalities:
1. \(3 + 6 > 9\) → \(9 > 9\) (not true)
2. \(3 + 9 > 6\) → \(12 > 6\) (true)
3. \(6 + 9 > 3\) → \(15 > 3\) (true)
Since the first condition is not satisfied (as \(9\) is not greater than \(9\)), the lengths 3, 6, and 9 cannot form a triangle. Therefore, the answer is **No**.
Damon
Try drawing it. You will see immediately that the three stuck on the end of the six will not reach the end of the 9
unless you make a straight line instead of a triangle.
unless you make a straight line instead of a triangle.
GPT-4o mini
You're absolutely correct! If you visualize or draw the side lengths 3, 6, and 9, you will see that connecting these lengths results in a straight line rather than forming a triangle. This is consistent with the triangle inequality theorem.
When the sum of the two shorter sides equals the length of the longest side (in this case, \(3 + 6 = 9\)), they don't form a triangle but rather a degenerate triangle, which is essentially a straight line. Thus, you cannot form a triangle with side lengths of 3, 6, and 9.
Thank you for your clarification!
When the sum of the two shorter sides equals the length of the longest side (in this case, \(3 + 6 = 9\)), they don't form a triangle but rather a degenerate triangle, which is essentially a straight line. Thus, you cannot form a triangle with side lengths of 3, 6, and 9.
Thank you for your clarification!