Question
Two sides of a triangle measure 4 and 6 which inequality shows all the possible lengths of the third side
Answers
GPT-4o mini
To find the possible lengths of the third side of a triangle when two sides are given, we can use the triangle inequality theorem. The triangle inequality states that the sum of the lengths of any two sides must be greater than the length of the third 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 the following conditions:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)
Substituting the values of \( a \) and \( b \):
1. \( 4 + 6 > c \) which simplifies to \( c < 10 \)
2. \( 4 + c > 6 \) which simplifies to \( c > 2 \)
3. \( 6 + c > 4 \) which simplifies to \( c > -2 \) (this condition is always true since \( c > 2 \))
From the inequalities derived, we can combine them to find:
\[ 2 < c < 10 \]
Thus, the possible lengths of the third side \( c \) must satisfy the inequality:
\[ 2 < c < 10 \]
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 the following conditions:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)
Substituting the values of \( a \) and \( b \):
1. \( 4 + 6 > c \) which simplifies to \( c < 10 \)
2. \( 4 + c > 6 \) which simplifies to \( c > 2 \)
3. \( 6 + c > 4 \) which simplifies to \( c > -2 \) (this condition is always true since \( c > 2 \))
From the inequalities derived, we can combine them to find:
\[ 2 < c < 10 \]
Thus, the possible lengths of the third side \( c \) must satisfy the inequality:
\[ 2 < c < 10 \]