Final answer:
The question is about determining the possible length of the third side of a triangle knowing the lengths of the other two sides, using the Triangle Inequality Theorem. Options a and d would result in valid triangles, thus the possible lengths for the third side are 12m and 10m.
Step-by-step explanation:
Joy is making a triangular flowerbed and wants to determine the length of the third side given two sides of lengths 5m and 13m. The Triangle Inequality Theorem states that in any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Based on the theorem, we can establish which of the provided options for the third side would create a valid triangle.
- For option a with a third side of 12m: 5m + 12m > 13m, which is true, and 5m + 13m > 12m, which is also true, and 12m + 13m > 5m, which is again true. So, a valid triangle can be formed.
- For option b with a third side of 8m: 5m + 8m > 13m, which is false, so a triangle cannot be formed.
- For option c with a third side of 18m: 5m + 13m > 18m, which is false, so a triangle cannot be formed.
- For option d with a third side of 10m: 5m + 10m > 13m, which is true, and 5m + 13m > 10m, also true, and 10m + 13m > 5m, again true. Thus, a valid triangle can be made.
Therefore, the possible lengths for the third side that would allow for a valid triangle are 12m and 10m, which are options a and d respectively.