Question
measure of two sides of a triangle is 7 and 11 what is the range for the third angle
Answers
GPT-4o mini
To determine the range for the third side of a triangle when the other two sides are given (7 and 11), you 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 the sides of the triangle be \( a = 7 \), \( b = 11 \), and \( c \) be the length of the third side.
The triangle inequalities are as follows:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)
Substituting the values of \( a \) and \( b \):
1. \( 7 + 11 > c \)
\( 18 > c \)
or \( c < 18 \)
2. \( 7 + c > 11 \)
\( c > 4 \)
3. \( 11 + c > 7 \)
This is always true since \( c \) is positive.
Combining these two inequalities, we get:
\[ 4 < c < 18 \]
Thus, the range for the length of the third side \( c \) is \( 4 < c < 18 \).
Let the sides of the triangle be \( a = 7 \), \( b = 11 \), and \( c \) be the length of the third side.
The triangle inequalities are as follows:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)
Substituting the values of \( a \) and \( b \):
1. \( 7 + 11 > c \)
\( 18 > c \)
or \( c < 18 \)
2. \( 7 + c > 11 \)
\( c > 4 \)
3. \( 11 + c > 7 \)
This is always true since \( c \) is positive.
Combining these two inequalities, we get:
\[ 4 < c < 18 \]
Thus, the range for the length of the third side \( c \) is \( 4 < c < 18 \).