Use the triangle inequality theorem to determine which three side lengths form a triangle
2,7,9
7,8,9
7,5,13
4,15,20
In order for three side lengths to form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
Let's apply the triangle inequality theorem to the given side lengths:
1. 2, 7, 9
- 2 + 7 > 9 (not true)
- 2 + 9 > 7 (true)
- 7 + 9 > 2 (true)
The side lengths 2, 7, 9 do not form a triangle.
2. 7, 8, 9
- 7 + 8 > 9 (true)
- 7 + 9 > 8 (true)
- 8 + 9 > 7 (true)
The side lengths 7, 8, 9 form a triangle.
3. 7, 5, 13
- 7 + 5 > 13 (not true)
- 7 + 13 > 5 (true)
- 5 + 13 > 7 (true)
The side lengths 7, 5, 13 do not form a triangle.
4. 4, 15, 20
- 4 + 15 > 20 (true)
- 4 + 20 > 15 (true)
- 15 + 20 > 4 (true)
The side lengths 4, 15, 20 form a triangle.
Therefore, only the side lengths 7, 8, 9 and 4, 15, 20 satisfy the triangle inequality theorem and form a triangle.