Question

The Triangle Inequality Theorem states that the sum of any two sides of a triangle must be greater than the measure of the third side.

a + b > c

Using this theorem, tell whether the examples below are possible triangles by selecting yes or no.

(4 points)
Yes No
a = 3, b = 2, c = 4


a = 7, b = 10, c = 18


a = 2, b = 4, c = 7


a = 5, b = 4, c = 7

Answer 1

Let's apply the Triangle Inequality Theorem \(a + b > c\), \(a + c > b\), and \(b + c > a\) to each set of side lengths:

1. For \(a = 3\), \(b = 2\), \(c = 4\):
- \(3 + 2 > 4\) (5 > 4, True)
- \(3 + 4 > 2\) (7 > 2, True)
- \(2 + 4 > 3\) (6 > 3, True)

Since all conditions are satisfied, the answer is **Yes**.

2. For \(a = 7\), \(b = 10\), \(c = 18\):
- \(7 + 10 > 18\) (17 > 18, False)
- \(7 + 18 > 10\) (25 > 10, True)
- \(10 + 18 > 7\) (28 > 7, True)

Since one condition is not satisfied, the answer is **No**.

3. For \(a = 2\), \(b = 4\), \(c = 7\):
- \(2 + 4 > 7\) (6 > 7, False)
- \(2 + 7 > 4\) (9 > 4, True)
- \(4 + 7 > 2\) (11 > 2, True)

Since one condition is not satisfied, the answer is **No**.

4. For \(a = 5\), \(b = 4\), \(c = 7\):
- \(5 + 4 > 7\) (9 > 7, True)
- \(5 + 7 > 4\) (12 > 4, True)
- \(4 + 7 > 5\) (11 > 5, True)

Since all conditions are satisfied, the answer is **Yes**.

In summary:
- \(a = 3, b = 2, c = 4\): **Yes**
- \(a = 7, b = 10, c = 18\): **No**
- \(a = 2, b = 4, c = 7\): **No**
- \(a = 5, b = 4, c = 7\): **Yes**