Final answer:
The length of the third side of a triangle with two known sides must be greater than the absolute difference of the given sides and less than their sum. For sides of 8 and 12 units, the third side must be greater than 4 units and less than 20 units. A general rule is that the third side must be more than |a - b| and less than a + b, where a and b are the lengths of the two sides.
Step-by-step explanation:
The question asks about the possible lengths of the third side of a triangle when two sides are known. In this case, the lengths of the sides are 8 and 12 units. To find the range within which the third side must lie, we use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
For a triangle with sides a and b, with b > a, the length of the third side c must be more than b - a and less than b + a. Therefore, for the given side lengths of 8 and 12:
This means the third side must be greater than 4 units and less than 20 units. The generalization for sides of lengths a and b would be:
- The third side must be greater than |a - b| (the absolute value of a - b).
- The third side must be less than a + b.