Question
two sides of a triangle is 10 and 12 what is the third length
Answers
GPT-4o mini
To determine the possible lengths of the third side of a triangle when the lengths of the other two sides are given (10 and 12), 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 lengths of the two sides be \( a = 10 \) and \( b = 12 \), and let the length of the third side be \( c \). According to the triangle inequality, we have the following conditions:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)
Substituting the known values into these inequalities:
1. \( 10 + 12 > c \) → \( c < 22 \)
2. \( 10 + c > 12 \) → \( c > 2 \)
3. \( 12 + c > 10 \) → \( c > -2 \) (This condition is always satisfied since \( c > 2 \))
Combining the valid inequalities, we find:
\[
2 < c < 22
\]
Therefore, the possible lengths for the third side \( c \) must be greater than 2 and less than 22. So, the third side can be any length in the range \( (2, 22) \).
Let the lengths of the two sides be \( a = 10 \) and \( b = 12 \), and let the length of the third side be \( c \). According to the triangle inequality, we have the following conditions:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)
Substituting the known values into these inequalities:
1. \( 10 + 12 > c \) → \( c < 22 \)
2. \( 10 + c > 12 \) → \( c > 2 \)
3. \( 12 + c > 10 \) → \( c > -2 \) (This condition is always satisfied since \( c > 2 \))
Combining the valid inequalities, we find:
\[
2 < c < 22
\]
Therefore, the possible lengths for the third side \( c \) must be greater than 2 and less than 22. So, the third side can be any length in the range \( (2, 22) \).