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
3.1
3
4
3.5
1
2.9

Answer 1

To determine which lengths can be the third side of a triangle with the other two sides measuring 7 inches and 4 inches, we can use the triangle inequality theorem. This theorem states that for any triangle with sides of lengths \(a\), \(b\), and \(c\), the following conditions must be satisfied:

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

Assigning the side lengths as follows:
- \(a = 7\)
- \(b = 4\)
- \(c\) is the unknown third side.

We now use the inequalities:

1. \(7 + 4 > c \Rightarrow 11 > c \Rightarrow c < 11\)
2. \(7 + c > 4 \Rightarrow c > 4 - 7 \Rightarrow c > -3\) (this condition will always be true since the side length must be positive)
3. \(4 + c > 7 \Rightarrow c > 3\)

From the valid conditions, we find:
- \(c > 3\)
- \(c < 11\)

Putting these together, the third side \(c\) must satisfy:
\[ 3 < c < 11 \]

Now, we can evaluate the given options:

- **3.1**: Valid, since \(3 < 3.1 < 11\).
- **3**: Not valid, since \(3\) is not greater than \(3\).
- **4**: Valid, since \(3 < 4 < 11\).
- **3.5**: Valid, since \(3 < 3.5 < 11\).
- **1**: Not valid, since \(1 < 3\).
- **2.9**: Not valid, since \(2.9 < 3\).

Now we can summarize:

**Could be the third side length**: 3.1, 4, 3.5
**Could not be the third side length**: 3, 1, 2.9