To determine which set of two sides is possible for the lengths of the other two sides alongside the 13 cm side in a triangle, we apply the triangle inequality theorem. This theorem states that the sum of any two sides of a triangle must be greater than the third side.
We will check each given pair of sides:
1. 5 cm and 8 cm:
- Check if [tex]\(13 + 5 > 8\)[/tex]: [tex]\(18 > 8\)[/tex] (True)
- Check if [tex]\(13 + 8 > 5\)[/tex]: [tex]\(21 > 5\)[/tex] (True)
- Check if [tex]\(5 + 8 > 13\)[/tex]: [tex]\(13 > 13\)[/tex] (False)
Since one condition is false, 5 cm and 8 cm cannot form a triangle with a side of 13 cm.
2. 6 cm and 7 cm:
- Check if [tex]\(13 + 6 > 7\)[/tex]: [tex]\(19 > 7\)[/tex] (True)
- Check if [tex]\(13 + 7 > 6\)[/tex]: [tex]\(20 > 6\)[/tex] (True)
- Check if [tex]\(6 + 7 > 13\)[/tex]: [tex]\(13 > 13\)[/tex] (False)
Since one condition is false, 6 cm and 7 cm cannot form a triangle with a side of 13 cm.
3. 7 cm and 2 cm:
- Check if [tex]\(13 + 7 > 2\)[/tex]: [tex]\(20 > 2\)[/tex] (True)
- Check if [tex]\(13 + 2 > 7\)[/tex]: [tex]\(15 > 7\)[/tex] (True)
- Check if [tex]\(7 + 2 > 13\)[/tex]: [tex]\(9 > 13\)[/tex] (False)
Since one condition is false, 7 cm and 2 cm cannot form a triangle with a side of 13 cm.
4. 8 cm and 9 cm:
- Check if [tex]\(13 + 8 > 9\)[/tex]: [tex]\(21 > 9\)[/tex] (True)
- Check if [tex]\(13 + 9 > 8\)[/tex]: [tex]\(22 > 8\)[/tex] (True)
- Check if [tex]\(8 + 9 > 13\)[/tex]: [tex]\(17 > 13\)[/tex] (True)
All conditions are true, so 8 cm and 9 cm can form a triangle with a side of 13 cm.
Therefore, the valid set of sides that can form a triangle with one side being 13 cm is:
8 cm and 9 cm
