Question

If a triangle has sides of 31 in. and 28 in., which is a possible length for the third side?
A. 33 in.
B. 62 in.
C. 59 in.
D. 3 in.

Answer 1

To determine a possible length for the third side of a triangle when two sides are given (31 in. and 28 in.), 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 length of the third side be \( x \). According to the triangle inequality theorem, we have three conditions:

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

Now we will calculate these inequalities:

1. \( 31 + 28 > x \)
\( 59 > x \)
or \( x < 59 \)

2. \( 31 + x > 28 \)
\( x > 28 - 31 \)
\( x > -3 \) (This is always true for positive lengths)

3. \( 28 + x > 31 \)
\( x > 31 - 28 \)
\( x > 3 \)

From the inequalities, we derive:

- \( x < 59 \)
- \( x > 3 \)

Combining these two results gives us:
\( 3 < x < 59 \)

Now we can evaluate the options provided:

A. 33 in. (valid, since \( 3 < 33 < 59 \))
B. 62 in. (invalid, since \( 62 > 59 \))
C. 59 in. (invalid, since \( 59 \text{ is not less than } 59 \))
D. 3 in. (invalid, since \( 3 \text{ is not greater than } 3 \))

Thus, the only possible length for the third side from the options given is **A. 33 in.**