To determine the possible lengths of the third side of a triangle given the lengths of the first two sides (10 and 12), we can use the triangle inequality theorem. This theorem states that for any triangle with sides a, b, and c, the following must be true:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Here, we have sides a = 10 and b = 12.
Let's denote the third side as c. According to the triangle inequality, we can derive the following inequalities:
- \( 10 + 12 > c \) → \( c < 22 \)
- \( 10 + c > 12 \) → \( c > 2 \)
- \( 12 + c > 10 \) → \( c > -2 \) (this is always true since c is a positive length)
From the first two inequalities, we can summarize that:
\( 2 < c < 22 \)
Now let's evaluate the provided options:
- 24: Not valid (c must be less than 22)
- 15: Valid (15 is in the interval 2 < c < 22)
- 30: Not valid (c must be less than 22)
- 22: Not valid (c must be strictly less than 22)
The only valid choice for the length of the third side is 15.