Question

If a triangle has sides of 31 in. and 28 in., which is a possible length for the third side?

33 inch
3 inch
62 inch
59 inch

Answer 1

To determine the possible lengths for the third side of a triangle given two sides, we can use the triangle inequality theorem. This theorem states that the sum of any two sides of a triangle must be greater than the third side.

Let the lengths of the two sides be \(a = 31\) inches and \(b = 28\) inches, and let the length of the third side be \(c\). The triangle inequality gives us the following three inequalities:

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

Substituting the values we have:

1. \(31 + 28 > c\)
\[
59 > c \quad \text{or} \quad c < 59
\]

2. \(31 + c > 28\)
\[
c > 28 - 31 \quad \text{or} \quad c > -3 \quad \text{(which means } c > 0\text{)}
\]

3. \(28 + c > 31\)
\[
c > 31 - 28 \quad \text{or} \quad c > 3
\]

Combining the results from the inequalities, we have:
\[
3 < c < 59
\]

So, the possible values for \(c\) must be greater than 3 inches and less than 59 inches. Now we can evaluate the provided options:

- \(33\) inches: this is within the range \( (3, 59) \)
- \(3\) inches: not valid as it is equal to the lower bound
- \(62\) inches: not valid as it exceeds the upper bound
- \(59\) inches: not valid as it is equal to the upper bound

Thus, the only valid option for the length of the third side is:

**33 inches**.