To determine which of the given sets of sides will form a triangle, we use the Triangle Inequality Theorem. This theorem states that for three sides to form a triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
Here are the steps for each set of side lengths:
1. Side [tex]\(a = 16 \, \text{cm}\)[/tex], side [tex]\(b = 8 \, \text{cm}\)[/tex], side [tex]\(c = 7 \, \text{cm}\)[/tex]:
- Check if [tex]\(a + b > c\)[/tex]: [tex]\(16 + 8 = 24\)[/tex], which is greater than [tex]\(7\)[/tex].
- Check if [tex]\(a + c > b\)[/tex]: [tex]\(16 + 7 = 23\)[/tex], which is greater than [tex]\(8\)[/tex].
- Check if [tex]\(b + c > a\)[/tex]: [tex]\(8 + 7 = 15\)[/tex], which is not greater than [tex]\(16\)[/tex].
Since [tex]\(b + c \leq a\)[/tex], these sides do not form a triangle.
2. Side [tex]\(a = 16 \, \text{cm}\)[/tex], side [tex]\(b = 9 \, \text{cm}\)[/tex], side [tex]\(c = 2 \, \text{cm}\)[/tex]:
- Check if [tex]\(a + b > c\)[/tex]: [tex]\(16 + 9 = 25\)[/tex], which is greater than [tex]\(2\)[/tex].
- Check if [tex]\(a + c > b\)[/tex]: [tex]\(16 + 2 = 18\)[/tex], which is greater than [tex]\(9\)[/tex].
- Check if [tex]\(b + c > a\)[/tex]: [tex]\(9 + 2 = 11\)[/tex], which is not greater than [tex]\(16\)[/tex].
Since [tex]\(b + c \leq a\)[/tex], these sides do not form a triangle.
3. Side [tex]\(a = 17 \, \text{cm}\)[/tex], side [tex]\(b = 12 \, \text{cm}\)[/tex], side [tex]\(c = 7 \, \text{cm}\)[/tex]:
- Check if [tex]\(a + b > c\)[/tex]: [tex]\(17 + 12 = 29\)[/tex], which is greater than [tex]\(7\)[/tex].
- Check if [tex]\(a + c > b\)[/tex]: [tex]\(17 + 7 = 24\)[/tex], which is greater than [tex]\(12\)[/tex].
- Check if [tex]\(b + c > a\)[/tex]: [tex]\(12 + 7 = 19\)[/tex], which is greater than [tex]\(17\)[/tex].
Since all conditions of the Triangle Inequality Theorem are satisfied, these sides form a triangle.
4. Side [tex]\(a = 17 \, \text{cm}\)[/tex], side [tex]\(b = 11 \, \text{cm}\)[/tex], side [tex]\(c = 5 \, \text{cm}\)[/tex]:
- Check if [tex]\(a + b > c\)[/tex]: [tex]\(17 + 11 = 28\)[/tex], which is greater than [tex]\(5\)[/tex].
- Check if [tex]\(a + c > b\)[/tex]: [tex]\(17 + 5 = 22\)[/tex], which is greater than [tex]\(11\)[/tex].
- Check if [tex]\(b + c > a\)[/tex]: [tex]\(11 + 5 = 16\)[/tex], which is not greater than [tex]\(17\)[/tex].
Since [tex]\(b + c \leq a\)[/tex], these sides do not form a triangle.
In summary, out of the four given possibilities, only the set with sides [tex]\(17 \, \text{cm}\)[/tex], [tex]\(12 \, \text{cm}\)[/tex], and [tex]\(7 \, \text{cm}\)[/tex] will form a triangle.
