Question

A triangle has side lengths of 7 inches and 4 inches. Which of the following could be the length of the third side in inches?
Could be the third side length Could not be the third side length
10.8
11
9.5
7
12
8

Answer 1

To determine the possible lengths of the third side of a triangle with side lengths of 7 inches and 4 inches, we can use the triangle inequality theorem. According to this theorem, for any triangle with sides \(a\), \(b\), and \(c\), the following conditions must hold:

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

Let \(c\) be the length of the third side. For the two known sides, we have \(a = 7\) and \(b = 4\).

We can apply the triangle inequality:

1. \(7 + 4 > c\) → \(11 > c\) → \(c < 11\)
2. \(7 + c > 4\) → \(c > -3\) (this condition is always true if \(c\) is a positive length)
3. \(4 + c > 7\) → \(c > 3\)

From these conditions, we conclude:

- The length of the third side \(c\) must be greater than 3 inches and less than 11 inches. Thus, the valid range for \(c\) is \(3 < c < 11\).

Now let's analyze the options provided:

- **10.8:** Valid (since \(3 < 10.8 < 11\))
- **11:** Not valid (since \(c\) must be less than 11)
- **9.5:** Valid (since \(3 < 9.5 < 11\))
- **7:** Valid (since \(3 < 7 < 11\))
- **12:** Not valid (since \(c\) must be less than 11)
- **8:** Valid (since \(3 < 8 < 11\))

**Conclusion:**
- **Could be the third side length:** 10.8, 9.5, 7, 8
- **Could not be the third side length:** 11, 12