To determine the length of the third side of a triangle with sides measuring 10 and 12, we can use the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's denote the lengths of the sides as follows:
- \( a = 10 \)
- \( b = 12 \)
- \( c \) = length of the third side
The triangle inequality gives us the following three conditions:
- \( a + b > c \) (10 + 12 > c)
- \( a + c > b \) (10 + c > 12)
- \( b + c > a \) (12 + c > 10)
Let's evaluate these inequalities:
-
From \( 10 + 12 > c \):
- \( 22 > c \)
- Thus, \( c < 22 \)
-
From \( 10 + c > 12 \):
- \( c > 2 \)
-
From \( 12 + c > 10 \):
- \( c > -2 \), which is always satisfied since \( c > 2 \).
Taking into account these inequalities, we find that:
- \( 2 < c < 22 \)
Now we can analyze the answer choices:
- 22: Not valid since \( c \) must be less than 22.
- 24: Not valid (out of range).
- 15: Valid choice (between 2 and 22).
- 30: Not valid (out of range).
Therefore, the valid length for the third side is 15.