Question

If a triangle has sides of 31 in. and 28 in., which is a possible length for the third side?(1 point)
Responses

33 in.
33 in.

59 in.
59 in.

62 in.
62 in.

3 in.

Answer 1

To determine the possible lengths for the third side of a triangle given two sides of lengths \( a \) and \( b \) (31 in. and 28 in., respectively), we can use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let the third side be \( c \). The triangle inequality gives us the following inequalities:

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

Let's apply these inequalities:

1. \( 31 + 28 > c \) ⇒ \( 59 > c \) or \( c < 59 \)
2. \( 31 + c > 28 \) ⇒ \( c > -3 \) (which is always true since side lengths are positive)
3. \( 28 + c > 31 \) ⇒ \( c > 3 \)

Combining these results, we find that \( 3 < c < 59 \).

Now, we can evaluate the options provided:

- 33 in. (valid)
- 59 in. (not valid; cannot be equal to 59)
- 62 in. (not valid; greater than 59)
- 3 in. (not valid; cannot be equal to 3)

Therefore, the only length that is a valid third side is **33 in.**