Final answer:
The length of side f in triangle DEF must be greater than 7 and less than 31, according to the triangle inequality theorem. Therefore, the correct answer is option B) 7 < f < 31.
Step-by-step explanation:
When determining the possible lengths for side f in triangle ΔDEF with sides d, e, and f, and given that d = 12 and e = 19, we must consider the triangle inequality theorem. This theorem states that the lengths of any two sides of a triangle must sum to more than the length of the third side. Applying this to our values for d and e, we find two inequalities: f + 12 > 19 and f + 19 > 12.
The first inequality simplifies to f > 7. The second inequality is always true since f will be a positive length and adding it to 19 will always result in a value greater than 12. However, we must also consider a third inequality from the theorem: 12 + 19 > f, which simplifies to f < 31. Therefore, considering both limits together, we find the range of possible lengths for side f must be greater than 7 and less than 31, which corresponds to option B) 7 < f < 31.