To determine if a triangle can be formed using a set of side measurements, we need to apply the triangle inequality theorem. According to the theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
Let's analyze each set of side measurements:
1. 12, 23, 7:
The sum of the lengths of the two shorter sides (12 and 7) is 19, which is greater than the length of the longest side (23). Similarly, the sum of the lengths of 12 and 23 is 35, which is greater than 7. Lastly, the sum of the lengths of 23 and 7 is 30, which is greater than 12. Therefore, a triangle can be formed with these side measurements.
2. 10, 7, 2:
In this case, the sum of the lengths of the two shorter sides (10 and 2) is 12, which is less than the length of the remaining side (7). Consequently, a triangle cannot be formed with these side measurements.
3. 8, 3, 11:
The sum of the lengths of the two shorter sides (8 and 3) is 11, which is equal to the length of the remaining side (11). According to the triangle inequality theorem, the sum of the lengths of any two sides must be greater than the length of the remaining side. Since this condition is not met, a triangle cannot be formed with these side measurements.
4. 5, 11, 8:
The sum of the lengths of the two shorter sides (5 and 8) is 13, which is greater than the length of the longest side (11). Additionally, the sum of the lengths of 5 and 11 is 16, which is also greater than 8. Finally, the sum of the lengths of 11 and 8 is 19, which is greater than 5. Hence, a triangle can be formed with these side measurements.
In summary, triangles can be formed using the side measurements of 12, 23, 7, and 5, 11, 8.