All four students correctly identified sets of side lengths that violate the Triangle Inequality Theorem.
The table shows the four students' responses:
| Student | Side lengths | Explanation
| Cadence | 2 cm, 3 cm, 6 cm | 2 + 3 = 5, and 5 < 6 |
| Enrique | 4 cm, 5 cm, 7 cm | 5 - 4 = 1, and 1 < 7 |
| Jocelyn | 5 cm, 6 cm, 9 cm | 9 - 6 = 3, and 5 > 3 |
| Myron | 2 cm, 8 cm, 8 cm | 2 + 8 = 10, and 10 > 8 |
All four students correctly identified sets of side lengths that will not make a triangle.
The Triangle Inequality Theorem states that the sum of any two side lengths of a triangle must be greater than the third side length. In other words, no side of a triangle can be longer than the sum of the other two sides.
This theorem can be used to determine whether or not a given set of side lengths will form a triangle. For example, in Cadence's example, the sum of the two shorter sides (2 cm and 3 cm) is 5 cm, which is less than the length of the longest side (6 cm). Therefore, these side lengths will not form a triangle.
Similarly, in Enrique's example, the difference between the two shorter sides (4 cm and 5 cm) is 1 cm, which is less than the length of the longest side (7 cm). Therefore, these side lengths will also not form a triangle.
Jocelyn's and Myron's examples work in a similar way. In Jocelyn's example, the difference between the two longest sides (5 cm and 9 cm) is 4 cm, which is greater than the length of the shortest side (6 cm). Therefore, these side lengths will form a triangle.
In Myron's example, the sum of the two shorter sides (2 cm and 8 cm) is 10 cm, which is greater than the length of the longest side (8 cm). Therefore, these side lengths will not form a triangle.
Therefore, all four students correctly identified sets of side lengths that will not make a triangle.