Question

Can the side lengths of a triangle be 3, 6 and 9? (1 point)

Answer 1

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.

Answer 2

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**.

Answer 3

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!