To determine the length of the third side of a triangle when the other two sides measure 10 and 12, we can use the triangle inequality theorem, which 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 lengths of the sides be \(a = 10\), \(b = 12\), and \(c\) (the length of the third side). According to the triangle inequality, we must satisfy the following conditions:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
Substituting the values of \(a\) and \(b\):
- \(10 + 12 > c\) simplifies to \(c < 22\)
- \(10 + c > 12\) simplifies to \(c > 2\)
- \(12 + c > 10\) simplifies to \(c > -2\) (this condition is always met since \(c\) is positive)
Thus, the length of the third side \(c\) must satisfy: \[ 2 < c < 22 \]
Now, let's evaluate the provided choices:
- 22 (not valid, since \(c\) must be less than 22)
- 30 (not valid, since \(c\) must be less than 22)
- 15 (valid, since \(2 < 15 < 22\))
- 24 (not valid, since \(c\) must be less than 22)
The only valid option is 15. Therefore, the length of the third side can be 15.
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