Question
two sides of a triangle is 4 and 6 what is the third length
Answers
GPT-4o mini
To find the possible lengths of the third side of a triangle when the other two sides are known, we can use the triangle inequality theorem. The theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
Let the lengths of the two sides be \(a = 4\) and \(b = 6\), and let the length of the third side be \(c\). According to the triangle inequality, we have:
1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)
Substituting the values into these inequalities:
1. \(4 + 6 > c \implies 10 > c \implies c < 10\)
2. \(4 + c > 6 \implies c > 2\)
3. \(6 + c > 4 \implies c > -2\) (this doesn't provide a new constraint since \(c > 2\) is stronger)
Combining the valid constraints:
- \(2 < c < 10\)
Thus, the length of the third side \(c\) must be greater than 2 and less than 10.
Let the lengths of the two sides be \(a = 4\) and \(b = 6\), and let the length of the third side be \(c\). According to the triangle inequality, we have:
1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)
Substituting the values into these inequalities:
1. \(4 + 6 > c \implies 10 > c \implies c < 10\)
2. \(4 + c > 6 \implies c > 2\)
3. \(6 + c > 4 \implies c > -2\) (this doesn't provide a new constraint since \(c > 2\) is stronger)
Combining the valid constraints:
- \(2 < c < 10\)
Thus, the length of the third side \(c\) must be greater than 2 and less than 10.