If a triangle has sides of 31 in. and 28 in., which is a possible length for the third side?(1 point) Responses 3 in. 3 in. 33 in. 33 in. 62 in. 62 in. 59 in.

Answer 1

To determine a possible length for the third side of a triangle with sides of lengths 31 inches and 28 inches, we can use the triangle inequality theorem. The 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 length of the third side be \( c \). According to the triangle inequality, we have the following conditions:

1. \( 31 + 28 > c \)
2. \( 31 + c > 28 \)
3. \( 28 + c > 31 \)

Now, let's analyze these inequalities:

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

Combining these results, we find that: \[ 3 < c < 59 \]

Now we can evaluate the given responses for a possible length of \( c \):

• 3 in.: not possible because \( c \) must be greater than 3.
• 33 in.: possible because \( 3 < 33 < 59 \).
• 62 in.: not possible because \( c \) must be less than 59.
• 59 in.: not possible because it cannot be equal to 59 (it must be less).

Thus, the only valid option from the responses is 33 in.