Question

Turn this into a short little paragraph:
Apply the triangle inequality theorem to each combination
The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
3 Check the first combination: {7, 9, 3}
7+9 > 3 (True)
7 + 3 > 9 (True)
9+ 3>7 (True)
Since all three inequalities are true, this combination can form a triangle.
4 Check the second combination: {7, 9, 10}
7 + 9 > 10 (True)
7 + 10 > 9 (True)
9 + 10 > 7 (True)
Since all three inequalities are true, this combination can form a triangle.
5 Check the third combination: {7, 3, 10}
7+3>10 (False, 10 > 10 is not true)
7 + 10 > 3 (True)
3 + 10 > 7 (True)
Since one inequality is false, this combination cannot form a triangle.
6
Check the fourth combination: {9, 3, 10}
9 + 3 > 10 (True)
9 + 10 > 3 (True)
3 + 10 > 9 (True)
Since all three inequalities are true, this combination can form a triangle.
Count the number of valid triangle combinations
The valid combinations are:
{7,9, 3}, {7, 9, 10}, {9, 3, 10}
There are 3 valid combinations.

Answer 1

Using the triangle inequality theorem, we can evaluate whether combinations of side lengths can form triangles. For the first combination {7, 9, 3}, all inequalities are satisfied: 7 + 9 > 3, 7 + 3 > 9, and 9 + 3 > 7, confirming it can form a triangle. The second combination {7, 9, 10} also holds true for all inequalities, allowing it to form a triangle. However, the third combination {7, 3, 10} fails the inequality 7 + 3 > 10, meaning it cannot form a triangle. Lastly, the combination {9, 3, 10} satisfies all inequalities, confirming it can form a triangle. In total, there are three valid combinations: {7, 9, 3}, {7, 9, 10}, and {9, 3, 10}.