Question

how many triangles can you make with the numbers 7 9 3 and 10

Answer 1

To determine how many triangles can be formed with the numbers 7, 9, 3, and 10 as side lengths, we need to check the triangle inequality theorem. The triangle inequality states that for any three sides \(a\), \(b\), and \(c\), the following must be true:

1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

We need to evaluate all combinations of three sides from the four numbers given (7, 9, 3, and 10). The combinations are:

1. \( (7, 9, 3) \)
2. \( (7, 9, 10) \)
3. \( (7, 3, 10) \)
4. \( (9, 3, 10) \)

Let's check each combination.

1. **For (7, 9, 3)**:
- \(7 + 9 = 16 > 3\) (true)
- \(7 + 3 = 10 > 9\) (false)
- Since one condition fails, this combination cannot form a triangle.

2. **For (7, 9, 10)**:
- \(7 + 9 = 16 > 10\) (true)
- \(7 + 10 = 17 > 9\) (true)
- \(9 + 10 = 19 > 7\) (true)
- All conditions are satisfied, so this combination can form a triangle.

3. **For (7, 3, 10)**:
- \(7 + 3 = 10 > 10\) (false)
- Since one condition fails, this combination cannot form a triangle.

4. **For (9, 3, 10)**:
- \(9 + 3 = 12 > 10\) (true)
- \(9 + 10 = 19 > 3\) (true)
- \(3 + 10 = 13 > 9\) (true)
- All conditions are satisfied, so this combination can form a triangle.

In conclusion, the pairs that can form triangles are (7, 9, 10) and (9, 3, 10).

Thus, the total number of triangles that can be formed is **2**.