Triangle Inequality Theorem Quick Check

Question

3 of 53 of 5 Items
Question
Apply the Triangle Inequality Theorem to determine whether a triangle can be formed from sides of lengths 7 millimeters, 8 millimeters, and 9 millimeters.(1 point)
Responses

No, because the sum of any two sides is greater than the third side.
No, because the sum of any two sides is greater than the third side.

No, because the sum of any two sides is less than the third side.
No, because the sum of any two sides is less than the third side.

Yes, because the sum of any two sides is less than the third side.
Yes, because the sum of any two sides is less than the third side.

Yes, because the sum of any two sides is greater than the third side.
Yes, because the sum of any two sides is greater than the third side.

Answer 1

To determine whether a triangle can be formed with sides of lengths 7 mm, 8 mm, and 9 mm, we can apply the Triangle Inequality Theorem. This theorem states that for any triangle with side lengths \(a\), \(b\), and \(c\), the following conditions must be satisfied:

1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

Let's check each condition with the provided side lengths:

1. \(7 + 8 > 9\) → \(15 > 9\) (True)
2. \(7 + 9 > 8\) → \(16 > 8\) (True)
3. \(8 + 9 > 7\) → \(17 > 7\) (True)

Since all three conditions are satisfied, a triangle can be formed. Therefore, the correct response is:

Yes, because the sum of any two sides is greater than the third side.